Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

PART 1. GENERAL THEORY
Cauchy's problem: existence theorems, uniqueness (local and global).
Higher order systems and equations: linear systems; equations and systems with constant coefficients.
Qualitative analysis for second order autonomous equations: phase plane; harmonic oscillators; solitary waves; perturbations (outlines).

PART 2.
Boundary value problems for second order equations: eigenvalues, eigenfunctions; Green function; non-linear boundary value problems (outline).
Stability: conservative systems; Lyapunov functions; linearization.

PART 3. POSSIBLE COMPLEMENTS
a) Introduction to Euler-Lagrange equations and calculation of variations.
b) Fourier series and transforms.
c) Introductions to Hamiltonian systems.

Ordinary Differential Equations and Dynamical Systems (Graduate Studies in Mathematics). AMS.
by Gerald Teschl (Author). ISBN-13: 978-0821883280

Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

The program focuses on major themes of Biological Evolution of animal and plant organisms.
Light and Life, Evolution of photomorphogenesis, Secondary metabolites of plants, The conquest of emerged lands, plant-insect co-evolution.

There are no specific textbooks. Students can refer to the material shown in the classroom and made available.
Recommended readings:

Taiz L., Zeiger E. - Fisiologia Vegetale - Piccin Editore

Rascio N. : Elementi di Fisiologia vegetale - EdiSES

PLANT PHYSIOLOGY (IN ENGLISH LANGUAGE)
LINCOLN TAIZ- EDUARDO ZEIGER
SINAUER ASSOCIATES (FIFTH EDITION)

Zocchi A.;Imparare ad Imparare; Amazon Kindle eBook/cartaceo copertina flessibile


The teacher receives every day (Monday-Friday) by appointment via institutional email

1. ATOMIC THEORY AND ATOMIC STRUCTURE. Atoms, molecules, moles; atomic and molecular weight. Atomic models: Rutheford, Bohr. Quantum theory, quantum numbers and energy levels. Polyelectronic atoms; periodic system.
2. CHEMICAL BONDS. Ionic bond. Covalent bond: and bonds. Polyatomic molecules: molecular structure. Hybridization and resonance. Molecular orbital. Metallic bond. Intermolecular forces.
3. NOMENCLATURE AND CHEMICAL REACTIONS. Oxides, hydroxides, acids, salts, ions. Balancing chemical reactions: redox reactions.
4. STATES OF AGGREGATION. Gas state, ideal gas law. Solid state: ionic, covalent, molecular and metallic solids. Conductors, semiconductors, insulators. Liquid and amorphous states. Phase transitions and phase diagrams.
5. SOLUTIONS. Concentration, colligative properties; electrolyte solutions.
6. THERMODYNAMICS. Matter, energy, heat, first and second principles; enthalpy, entropy, free energy.
7. CHEMICAL EQUILIBRIUM. Equilibrium constant and free energy. Gas-phase and heterogeneous equilibria. Le Chatelier’s principle. Van’t Hoff equation.
8. EQUILIBRIA IN SOLUTION. Acid-base equilibria: acids and bases, pH, dissociation constant, polyprotic acids, hydrolysis, buffers. Acid-base titrations and pH indicators. Solubility equilibria: solubility product, common ion effect.
9. ELECTROCHEMISTRY. Batteries, electrode potentials, Nernst’s equation. Electrolysis.
10. CHEMICAL KINETICS. Reaction speed, speed constant. Influence of the temperature on the reaction speed: Arrhenius equation. Catalysts.
Numerical exercises on all the listed subjects.

P.W.Atkins, L. Jones; CHEMISTRY: MOLECULES, MATTER, AND CHANGE

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

Short introduction to Julia for scientific programming. Introduction to parallel aechitectures. Designi principles of parallel algorithms. Prallel and distributed programming with Julia. Comunication and sincronization primitives: MPI paradigm. Directive-based languages: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: mentions to BLAS, LAPACK, scaLAPACK. Sparse linear systems. Mentions to CombBLAS and GraphBLAS.

1. [Lecture slides and diary](https://github.com/cvdlab-courses/pdc/blob/master/schedule.md)

2. [Learning Julia](https://www.manning.com/books/julia-in-action)

3. Blaise N. Barney, [HPC Training Materials](https://computing.llnl.gov/tutorials/parallel_comp/), per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

4. J. Dongarra, J. Kurzak, J. Demmel, M. Heroux, [Linear Algebra Libraries for High- Performance Computing: Scientific Computing with Multicore and Accelerators](http://www.netlib.org/utk/people/JackDongarra/SLIDES/sc2011-tutorial.pdf), SuperComputing 2011 (SC11)

Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

1. Introduction. We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim is to prove
Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using the study of geodesics; namely
the curves which, at least locally, minimize the distance on a Riemannian manifold.
2. Integration on surfaces. Area of a surface and total Gaussian curvature.
3. Covariant derivative. Covariant derivative of a tangent vector field. Parallel transport and
geodesics. Geodesic curvature.
4. Gauss-Bonnet theorem. Proof of Gauss-Bonnet theorem, local and global version. Relations between topology and geometry of surfaces.
5. Hopf-Rinow theorem. Riemnnian manifolds of arbitrary dimension. The exponential map in Riemannian geometry. Convex neighborhoods. Complete
manifolds: proof of Hopf-Rinow theorem. Applications: rigidity of the sphere.
6. Exercises. Written exercises at home and in class.

[1] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976). [2] M. Do Carmo , Riemannian Geometry. Birk ̈auser, (1992).
[3] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
[4] Marco Abate, Francesca Tovena, Geometria Differenziale. Springer, (2011).

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Ω)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Ω)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Analisi funzionale, H. Brézis, Liguori Editore

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

Classification of the semilinear partial differential equations in any dimension. Classification in 2 dimensions and canonical forms. Wave equation in an interval using the separation of variables method: homogeneous case, general case. Wave equation on the line: D’Alambert’s solution. The half line case. Wave equation in R3.
Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc- tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa- tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.

A.N. Tichonov, A.A. Samarskij Equazioni della Fisica Matematica Edizioni MIR

Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Classification of the semilinear partial differential equations in any dimension. Classification in 2 dimensions and canonical forms. Wave equation in an interval using the separation of variables method: homogeneous case, general case. Wave equation on the line: D’Alambert’s solution. The half line case. Wave equation in R3.
Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc- tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa- tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.

A.N. Tichonov, A.A. Samarskij Equazioni della Fisica Matematica Edizioni MIR

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Physical quantities. Intensive and extensive physical quantities. Direct and indirect measurements.
Basic and derived quantities.
Units of measurement. Units of measurement systems. Change of units. Dimensions,
physical principle of homogeneity and dimensional analysis.
Measurement tools. Analogical and Digital Instruments.
Characteristics of the instruments: Range, Threshold, Resolution, Linearity and Sensitivity. Accuracy and Precision.
Uncertainty in measurements. Definition of measurement error. Random errors and
systematic errors. Concept of measurement uncertainty. Causes of uncertainties. Uncertainties of Type
A and Type B.
Graphical analysis of data. Usage of tables and graphs for representation and preliminary analysis
of data without the use of statistical tools. Linear, semi-logarithmic graphs,
Double-logarithmic. Histograms.
Propagation of uncertainties. Uncertainty in indirect measurements. Propagation
of uncertainties for independent quantities. Correlated random variables. Definition of
correlation coefficient. Propagation of uncertainties for correlated quantities.


Laboratory program
- Measurements of fundamental quantities: mass, length, time
- Determination of measurement uncertainty: sensitivity of the instrument,
- Standard deviation in repeated measurements, propagation of uncertainties
- Uncertainty on the average in repeated measurements and dependence on sample size
- Study of the pendulum: verification of the independence of the period from the mass, study of the dependence of the period on the length, measurement of g
- Study of the motion of a cart on the inclined plane, effect of friction, measurement of g
- Static and dynamic study of the elastic constant of a spring
- Measurement of resistances with voltamperometric method, study of a resistive voltage divider
- Study of diffraction, verification of Snell's law

notes distributed during the classes

Physical quantities. Intensive and extensive physical quantities. Direct and indirect measurements.
Basic and derived quantities.
Units of measurement. Units of measurement systems. Change of units. Dimensions,
physical principle of homogeneity and dimensional analysis.
Measurement tools. Analogical and Digital Instruments. Characteristics of the instruments: Range, Threshold, Resolution, Linearity and Sensitivity. Accuracy and Precision.
Uncertainty in measurements. Definition of measurement error. Random errors and
systematic errors. Concept of measurement uncertainty. Causes of uncertainties. Uncertainties of Type
A and Type B.
Graphical analysis of data. Usage of tables and graphs for representation and preliminary analysis
of data without the use of statistical tools. Linear, semi-logarithmic graphs,
Double-logarithmic. Histograms.
Propagation of uncertainties. Uncertainty in indirect measurements. Propagation
of uncertainties for independent quantities. Correlated random variables. Definition of
correlation coefficient. Propagation of uncertainties for correlated quantities.

Laboratory program
- Measurements of fundamental quantities: mass, length, time
- Determination of measurement uncertainty: sensitivity of the instrument,
- Standard deviation in repeated measurements, propagation of uncertainties
- Uncertainty on the average in repeated measurements and dependence on sample size
- Study of the pendulum: verification of the independence of the period from the mass,
study of the dependence of the period on the length, measurement of g
- Study of the motion of a cart on the inclined plane, effect of friction, measurement of g
- Static and dynamic study of the elastic constant of a spring
- Measurement of resistances with voltamperometric method, study of a resistive voltage divider
- Study of diffraction, verification of Snell's law

notes distributed during the classes

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense
D.M. Burton, Elementary Number theory, McGraw-Hill, (sesta edizione 2007)
G.A. Jones e J.M. Jones, Elementary Number theory, Springer, 1998 (ristampa 2008)
H. Davenport, Aritmetica superiore. Una introduzione alla teoria dei numeri, Zanichelli, (1994)
G.H. Hardy e E.M. Wright, An introduction to the theory of numbers, Oxford University Press, (1979)
K.H. Rosen, Elementary number theory and its applications, Addison-Wesley (2000)

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Introductory notions
Recap of Special Relativity. Lorentz transformations in Minkowski’s space. Vectors in Minkowski’s space. Basis of the tangent space.
Cotangents space and dual vectors in Minkowski’s space. Basis of cotangent space. Lorentz transformations of vectors and dual vectors. Tensors in Minkowski’s space. Properties of vectors, dual vectors and tensors in Minkowski’s space. Definition of symmetric and antisymmetric tensor. Symmetrization and antisymmetrization of a generic tensor. Metric in Minkowski’s space: definition and properties. Operations related to the metric: scalar products, rising and lowering indices of a tensor, contractions and trace of a tensor. Equivalence between inertial and gravitational mass. Weak Equivalence Principle (WEP), Einstein’s equivalence Principle (EEP).

Basic notions of differential geometry
Introduction to the notion of manifold. Definition and properties of maps. Injective and suriective maps (some examples included). Composition of charts. Invertible charts. Definition of diffeomorphism.
Definition of chart (or coordinate system). Definition of atlas. Definition of manifold. Product of manifolds.
Formal coordinate independent definition of vector. Demonstration that the dimension of the tangent space coincides with the one of the corresponding manifold. Basis (or coordinate system) of the tangent space. Coordinate transformations. Coordinate transformations of the components of a vector. Definition and properties of the tangent field. Definition of one parameter group of diffeomorphisms. Definition of integral curves. Commutator of two vectors. Coordinate independent definition of dual vector (one-form). Cotangent space and corresponding basis. Coordinate transformation of the components of a one-form. Coordinate independent definition of tensor. Demonstration that the partial derivative of a tensor is not a tensor.
Metric: signature and canonical form.
Tensor densities.
Differential forms. Wedge product. Exterior derivative. Closed and exact form. Poincarre Lemma (statement only). Hodge duality. Maxwell equations expressed in term of exterior derivative and hodge duality (only small reference).
Integration over a manifold: volume element in terms of the determinant of the metric.
Maps between manifold: pullback and pushforward. Pullback and pushforward associated to diffeomorphisms. Equivalence between diffeomorphisms and coordinate transformations. Vector field associated to diffeomorphisms. Lie Derivative: definition and general properties. Action of Lie’s derivative on scalars, vectors, one-forms and tensors. General Relativity as diffeomorphism invariant theory. Analogy between gauge transformations and diffeomorphisms.

Symmetries.
Notion of submanifold. Immersed and embedded submanifolds. Notion of hypersurface and boundary of a manifold.
Integration on manifolds again: differential form as generic volume element. Orientation and orientable manifold. Covering of the manifold through partition of unity. Integration of p-forms over submanifold. Demonstrations that the volume element can be expressed in terms of the determinant of the metric. Stokes theorem (no demonstration).


Connection, Covariant Derivative, Curvature
Lie’s Algebra and Lie’s group. Action from the right and from the left. Left- and right-invariant vectors. Structure constants. Examples of Lie groups. Maurer-Cartan forms. Maurer-Cartan’s equations. Action of Lie Groups on manifolds. Definition of free, effective and transitive action. Orbit and stabilizer.
Algebric definition of connection and covariant derivative. General properties of covariant derivatives. Action of coordinate transformations on the connection.
Demonstration that the difference of Christoffel coefficients associated to two different connections transforms as a tensor; torsion tensor, torsion-free and metric connection. Demonstrations that for any given metric exists a connection (metric connections) for which the covariant derivative of the metric is zero.
Formal construction of the covariant derivative from the notion of parallel transport (qualitative introduction).
Fiber bundle. Trivial and locally trivializable bundles. Local trivilizations. Maps between fiber bundles (notions).
Defintion of bundle atlas, G-atlas, G-structure. Fiber Bundle with structure group G. Definition of Principal Bundle. Definition of section of a bundle. Vector bundle and bundle of basis, definition and general properties. Relation between principle bundle, vectorbundle and bundle of frames (definition of associated vector bundle to a principal bundle.
Construction of the covariante derivative on a vectorbundle (only the knowledge of the fundamental logical steps is required for the exam).
Curvature tensor as 2-form on a fiber bundle. Geometrical interpretation of the curvature. Bianchi identity. Fiber metric. Ortogonal basis.
Connections and gauge theories: electromagnetism as simple example.
Soldering form. Choice of the gauge. Ortonormal and metric gauge.
Levi-Civita connection; Riemann’s tensor : definition and properties. Ricci’s tensor and scalar, Weyl’s tensor. Globally and locally inertial coordinates.

Einstein’s theory of gravity

Minimal coupling.
Particle in a gravitational field: affine parameter, self-parallel curves. Geodesic’s equations. Geodesic deviation.
Derivation of the Einstein’s equations from Newton’s limit.
Lagrangian derivations of Einstein’s equations.
General considerations on the structure of Einstein’s equations. Choice of the gauge. Energy conditions.
Symmetries and Killing vectors: version of Noether’s theorem from general relativity. Maximal number of linearly independent Killing vectors on a manifold. Homogenous and isotropic manifold. Spaces at constant curvature. Metric in spaces at constant curvature.

Notable solutions of Einstein’s equations

Static spherically symmetric spacetimes. Determination of Schwarzschild’s metric.
Cosmological solution. Spatially homogeneous and isotropic spacetime. Frieman’s Robertson-Walker metric. Friedman’s equations.
Coordinate singularities. Case of study: Schwarzschild radius. Rindler metric. Kruskal coordinates. Black hole solution.
Perturbation around a background metric. Case of study:perturbation of flat metric. Degrees of freedom. Linearized Einstein’s equations. Choice of the gauge. Linearized Einstein’s equations in vacuum: gravitational waves. Solutions in presence of the source (only few words).


Advanced concepts
Conformal transformations. Cotton’s tensor. Conformally flat metric. Demonstration of the theorem: a metric is conformally flat if and only if Weyl (Cotton) tensor is null. Conformal group. Conformal Killing vectors.
Alternative theories of gravity. Scalar-tensor theories. Jordan and Einstein’s frames.

1. S. Carrol Space time and Geometry: An Introduction to General Relativity (Addison Wesley, 2004);
2. R. Wald General Relativity (The Chicago Press, 1984);
3. B. Schutz A First Course in General Relativity (Cambridge Press)
4. people.sissa.it/~percacci/lectures/general/index.html
5. B. Schutz Geometrical Methods of Mathematical Physics (Cambridge Press)
6. S. Weinberg Gravitation and Cosmology-principles and application of the general theory of relativity (John Weiley & Sons, 1972);

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

Sheaf theory and its use in on schemes

Preseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.

Cohomology of sheaves

Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.

Cohomology of quasi-coherent and coherent sheaves on a scheme.

Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.

Invertible sheaves and linear systems

Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems. Base points. Linear systems, ample
and very ample sheaves. Amplitude criterion.

Notes from Prof. Sernesi
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).

Notations of analytic number theory. Euler's constant, the Dirichlet problem for the average number of divisors of an integer, the hyperbole method.

Chebichev theorems. Mertens theorem.

Dirichlet theorem first in arithmetic progression. The real function ζ (s), The Dirichlet theorem first in arithmetic progression, L-series of Dirichlet L (s, χ), productive.

Gaussian sums, Poisson sum formula, application to Gaussian sums. Dirichlet characters, determination of explicit characters,
orthogonality laws of characters, Dirichlet's theorem in the general case.

Analytical extension to s 0 of the function ζ (s) and of the Dirichlet L-series. Demonstration of de la Valleé Poussin that the L-series does not cancel each other out in s = 1. Mertens theorem first in arithmetic progression.

The Riemann function ζ. The Riemann article and the analytical extension of ζ (s). Riemann program for the proof of the prime number theorem. Riemann proof of the functional equation for ζ (s), trivial zeros for ζ (s). Hadamard products. Whole functions of finite order. Hadamard theorem for integer functions of order one. Distribution of the zeros of whole functions of order one. The ordinate of the non-trivial zero smaller than ζ is in absolute value greater than 6.5.

Logarithmic derivatives of the function ξ (s). The series of reciprocals of zeros in the critical strip. The zeta function has no zeros on the line Re(s) = 1. The Gamma function. Region free of zeros for ζ (Hadamard theorem - de La Valleé Poussin 1896). Von Mangoldt's formula for N (T).

Distribution of the first. The explicit formula for the function ψ (x), the discontinuous integral of Perron. The prime number theorem. Consequences of the Riemann hypothesis.

Davenport, Harold, Multiplicative number theory. Graduate Texts in Mathematics, 74.Springer-Verlag, New York, (2000).

Tenenbaum, G erald, Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, (1995).

The course is based on a set of single-topic lessons that will address the following topics:
From the computer to the information system: evolution of IT systems and information systems;
Introduction to enterprise information systems: the context in which they are inserted, the needs for which a complex information system is designed and implemented, the main components of the system;
Operating systems: the main components of an operating system and the aspects that are delegated to this basic software;
Computer networks: the support technology for communication between computers, the TCP / IP communication protocol and the highest level application protocols;
Relational databases: information archives, relational model, SQL language, DBMS tools;
Data warehouse: models and tools for the management and use of large “informational” data stores;
Web based applications: basic concepts, languages and protocols and architectures for the development of web based applications;
Information system security: the structural and organizational aspects that must be addressed, also through specific hardware and software tools, to guarantee the confidentiality, availability and integrity of information;
Principles of Software Engineering: some notes on standards, methodologies, best practices to ensure the quality and correct engineering of software products, information systems and IT services.

M. Pighin, A. Marzona, Sistemi Informativi Aziendali - Struttura e applicazioni, seconda edizione, Pearson, 2011.

Lecture notes by the teacher published on the course website (http://www.mat.uniroma3.it/users/liverani/IN530).

CONTENTS OF THE LECTURES: the numbers in round brackets refer to the chapter and section of the textbook adopted.
Kinetic theory of gases. Boltzmann equation and H theorem. (1, Par.2.1,2.2,2.3,2.4)
Maxwell-Boltzmann distribution. (1, Par. 2.5)
Phase space and Liouville theorem. (1, Par. 3.1,3.2)
Gibbs ensembles. Micro canonical ensemble. Definition of entropy. (1, Par. 3.3,3.4)
The ideal gas in the micro canonical ensemble. (1, Par. 3.6)
The equipartition theorem. (1, Par. 3.5)
The canonical ensemble. (1, Par.4.1).
The partition function and the free energy. Fluctuations of energy in the canonical ensemble. (1 Par. 4.4)
The grand canonical ensemble. The grand potential. The ideal gas in the grand canonical ensemble. (1 Par. 4.3).
Fluctuations of the particle number. (1 Par. 4.4)
Classical theory of the linear response and fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein quantum statistics. ( 1, Par. 7.1)
The Fermi gas. The Sommerfeld expansion and the electron specific heat. (1, Par. 7.2)
The Bose gas. The Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)


Web page with additional material about the lecture course

https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi

Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.

Further reading:
2) K. Huang, Meccanica Statistica, Zanichelli, 1997.
3) L. Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri, 2003.
4) Joel L. Lebowitz, Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics, 71, S346 (1999).

Further information is available on
https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi

The Course aims to introduce, in an elementary way and building on algebraic and geometric tools offered to the student in previous courses, those algebraic varieties which admit rational parametric equations. A sequel of relevant examples, both from the geometrical and historical points of view, will be useful to describe the contemporary state of the art on rational parametric equations and, more specifically, on what is known, or not known, on the rationality problem for some typical classes of algebraic varieties.
The Course will consider the following topics:
(1) Curves and rationality: Lueroth problem.
(2) Embeddings of the projective line.
(3) Projective varieties of minimal degree.
(4) Del Pezzo surfaces.
(5) Rationality criterion for algebraic surfaces.
(6) (Uni)rationality problems for hypersurfaces.
(7) Lueroth problem.
(8) Cubics.
(9) Artin-Mumford's counterexample.
(10) Outline of Vosin's method.

Text;
A. Corti, J. Kollar, K. Smith 'Rational and nearly rational varieties', Cambridge studies in advanced mathematics vol. 92, Cambridge UP (2004)

Some reference books for the course:
1) A. Beauville, B. Hassett, A. Kuznetsov, A. Verra 'Rationality problems in Algebraic Geometry' Lecture Notes in Mathematics vol. 2172, Springer (2016)
2) I. Dolgachev 'Classical Algebraic Geometry. A Modern View' Cambridge UP (2016)
3) M. Mustata 'Topics in Algebraic Geometry II. Rationality of Algebraic Varieties', Note on line: https://web.math.princeton.edu/~takumim/Rationality.pdf

A) STRUCTURAL RULES EXPRESSED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC, LUDICS
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

On line lecture notes

Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3,3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

Modulo A:

Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.

Part 2: Provability and satisyability.
First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.

Part 3: Towards proof-theory: the cut-elimination theorem.
The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.

Modulo B: Towards model theory: some consequences of the compactness theorem

Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. Notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITE' ET DECIDABILITE'. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3,3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

Modulo A:

Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.

Part 2: Provability and satisyability.
First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.

Part 3: Towards proof-theory: the cut-elimination theorem.
The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.

Modulo B: Towards model theory: some consequences of the compactness theorem

Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

Modulo A:

Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.

Part 2: Provability and satisyability.
First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.

Part 3: Towards proof-theory: the cut-elimination theorem.
The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.

Modulo B: Towards model theory: some consequences of the compactness theorem

Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. Notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITE' ET DECIDABILITE'. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

1. Classic Cryptography

- Basic cryptosystems: encryption by substitution, by translation, by permutation, affine cryptosystem, by Vigenère, by Hill. Stream encryption (synchronous and asynchronous), Linear feedback shift registers (LFSR) on finite fields, Autokey cipher. Product ciphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine ciphers, for substitution cipher (frequency analysis), for Vigenere cipher: Kasiski test, coincidence index; cryptoanalysis of Hill's cipher and LFSR: algebraic attacks, cube attack.

2. Application of Shannon theory to cryptography

- Security of ciphers: computational security, provable security, unconditional security. Basics of probability: discrete random variables, joint probability, conditional probability, independent random variables, Bayes' theorem. Random variables associated with cryptosystems. Perfect secrecy for encryption systems. Vernam cryptosystem. Entropy. Huffman codes. Spurious Keys and Unicity distance.

3. Block ciphers

- iterative encryption schemes; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; Differential cryptanalysis for SPN; Feistel ciphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the mcd and inverse; Operating modes for block ciphers.

4. Hash functions and codes for message authentication

- Hash functions and data integrity. Safe hash functions: resistance to the pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).

[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press.
[2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC.
[3] Delfs H., Knebl H., Introduction to Cryptography, (2007) Springer Verlag.

Short introduction to Julia for scientific programming. Introduction to parallel aechitectures. Designi principles of parallel algorithms. Prallel and distributed programming with Julia. Comunication and sincronization primitives: MPI paradigm. Directive-based languages: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: mentions to BLAS, LAPACK, scaLAPACK. Sparse linear systems. Mentions to CombBLAS and GraphBLAS.

1. [Lecture slides and diary](https://github.com/cvdlab-courses/pdc/blob/master/schedule.md)

2. [Learning Julia](https://www.manning.com/books/julia-in-action)

3. Blaise N. Barney, [HPC Training Materials](https://computing.llnl.gov/tutorials/parallel_comp/), per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

4. J. Dongarra, J. Kurzak, J. Demmel, M. Heroux, [Linear Algebra Libraries for High- Performance Computing: Scientific Computing with Multicore and Accelerators](http://www.netlib.org/utk/people/JackDongarra/SLIDES/sc2011-tutorial.pdf), SuperComputing 2011 (SC11)

The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification.
Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine.
Understanding the Object Oriented paradigm together with other non imperative approaches.

[1] Maurizio Gabbrielli, Simone Martini,Programming Languages - Principles and paradigms, 2/ed. McGraw-Hill, (2011).
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014).
[3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer- Verlag, (2012).
Course Slides

The course is based on a set of single-topic lessons that will address the following topics:
From the computer to the information system: evolution of IT systems and information systems;
Introduction to enterprise information systems: the context in which they are inserted, the needs for which a complex information system is designed and implemented, the main components of the system;
Operating systems: the main components of an operating system and the aspects that are delegated to this basic software;
Computer networks: the support technology for communication between computers, the TCP / IP communication protocol and the highest level application protocols;
Relational databases: information archives, relational model, SQL language, DBMS tools;
Data warehouse: models and tools for the management and use of large “informational” data stores;
Web based applications: basic concepts, languages and protocols and architectures for the development of web based applications;
Information system security: the structural and organizational aspects that must be addressed, also through specific hardware and software tools, to guarantee the confidentiality, availability and integrity of information;
Principles of Software Engineering: some notes on standards, methodologies, best practices to ensure the quality and correct engineering of software products, information systems and IT services.

M. Pighin, A. Marzona, Sistemi Informativi Aziendali - Struttura e applicazioni, seconda edizione, Pearson, 2011.

Lecture notes by the teacher published on the course website (http://www.mat.uniroma3.it/users/liverani/IN530)

1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.

R. Trudeau: "La rivoluzione non Euclidea" Bollati Boringhieri
V, Nikulin, I. Shafarevich "Geometries and groups" Springer ed.

The program focuses on major themes of Biological Evolution of animal and plant organisms.
Light and Life, Evolution of photomorphogenesis, Secondary metabolites of plants, The conquest of emerged lands, plant-insect co-evolution.

There are no specific textbooks. Students can refer to the material shown in the classroom and made available.
Recommended readings:
Taiz L., Zeiger E. - Fisiologia Vegetale - Piccin Editore

Rascio N. : Elementi di Fisiologia vegetale - EdiSES

PLANT PHYSIOLOGY (IN ENGLISH LANGUAGE)
LINCOLN TAIZ- EDUARDO ZEIGER
SINAUER ASSOCIATES (FIFTH EDITION)

Zocchi A.;Imparare ad Imparare; Amazon Kindle eBook/cartaceo copertina flessibile

The teacher receives every day (Monday-Friday) by appointment via institutional email

The environment of celestial bodies; the Solar System; the shape and size of the Earth; the main motions of the Earth (rotation and revolution); the long-term movements of the Earth; the Earth-Moon system.
Orientation; time and its measure; the representation of the earth's surface; the geographical coordinates (latitude and longitude), the reading of topographic maps.
The shapes of the relief and the modeling of the terrestrial relief; the marine hydrosphere, the continental hydrosphere, the cryosphere; the atmosphere; the climate.
The materials of the Earth: the minerals, the lithogenetic processes, the lithogenetic cycle.
The evolution of the Earth.

Capire la Terra
J.P. Grotzinger, T-H Jordan
(Terza edizione italiana condotta sulla settima edizione americana)

Il Globo Terrestre e la sua evoluzione
E. L. Palmieri e M. Parotto
Sesta Edizione (2008)

Educational material distributed during the course

1. ATOMIC THEORY AND ATOMIC STRUCTURE. Atoms, molecules, moles; atomic and molecular weight. Atomic models: Rutheford, Bohr. Quantum theory, quantum numbers and energy levels. Polyelectronic atoms; periodic system.
2. CHEMICAL BONDS. Ionic bond. Covalent bond: and bonds. Polyatomic molecules: molecular structure. Hybridization and resonance. Molecular orbital. Metallic bond. Intermolecular forces.
3. NOMENCLATURE AND CHEMICAL REACTIONS. Oxides, hydroxides, acids, salts, ions. Balancing chemical reactions: redox reactions.
4. STATES OF AGGREGATION. Gas state, ideal gas law. Solid state: ionic, covalent, molecular and metallic solids. Conductors, semiconductors, insulators. Liquid and amorphous states. Phase transitions and phase diagrams.
5. SOLUTIONS. Concentration, colligative properties; electrolyte solutions.
6. THERMODYNAMICS. Matter, energy, heat, first and second principles; enthalpy, entropy, free energy.
7. CHEMICAL EQUILIBRIUM. Equilibrium constant and free energy. Gas-phase and heterogeneous equilibria. Le Chatelier’s principle. Van’t Hoff equation.
8. EQUILIBRIA IN SOLUTION. Acid-base equilibria: acids and bases, pH, dissociation constant, polyprotic acids, hydrolysis, buffers. Acid-base titrations and pH indicators. Solubility equilibria: solubility product, common ion effect.
9. ELECTROCHEMISTRY. Batteries, electrode potentials, Nernst’s equation. Electrolysis.
10. CHEMICAL KINETICS. Reaction speed, speed constant. Influence of the temperature on the reaction speed: Arrhenius equation. Catalysts.
Numerical exercises on all the listed subjects.

P.W.Atkins, L. Jones; CHEMISTRY: MOLECULES, MATTER, AND CHANGE

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Fermat's last Theorem for regular primes
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Complex numbers: algebraic and topological properties. Geometric representation of complex numbers: polar coordinates and the complex exponential.
Complex functions with complex variables: continuity and properties, differentiability and first properties. Holomorphic function: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic function are harmonic conjugated.
Equations of Cauchy-Riemann: proof. Examples. Sequences and complex series. Properties. Power series with complex values. Abel's theorem and Hadamard's formula.
Proof of Abel's Theorem. Taylor's formula for series of complex powers. The exponential and the trigonometric functions as analytical functions. Basic properties.
Periodicity of the complex exponential function. The complex logarithm: first considerations. The ring of formal powers series with complex coefficients: basic properties. Analytical functions: definition and first properties.
Series of converging powers are analytical within the convergence region. Composition of analytical functions. Theorem of the inverse function.
Inverse by composition of a formal series and its convergence. Complex powers and properties. The binomial series and properties. Consequences of the inverse theorem: the canonical form of an analytic function.
Local properties of analytical function: open function theorem, invertibility criterion, principle of the maximum local module. The fundamental theorem of algebra. Parameterized curves. A holomorphic function with zero derivative is constant.
The place of the zeros of a non-constant analytical function is discrete. Analytical continuation of functions defined on open connected sets. Principle of the maximum global module. Integrals in paths: definition and first properties. Examples.
A continuous function in a connected open admits a primitive if and only if its integral along a closed curve is zero. Integration of uniformly converging series of functions. Examples. Local primitive of a holomorphic function.
Local primitive of a holomorphic function. The Goursat theorem. Integral of a holomorphic function along a continuous path.
The homotopical form of the Cauchy Theorem. Global primitive of a holomorphic function in a simply connected domain. Applications to the study of the logarithm.
The integral formula of Cauchy. Cauchy formula for development in series and applications: a holomorphic and analytical function; the theorem of Liouville and the fundamental theorem of algebra.
Integral formula for derivatives. The number of windings of a curve with respect to a point. Curves homologous to 0. The global formula of Cauchy.
Demonstration of the global Cauchy formula. Examples.
The first homology group of an open set with values ​​in integers. The Cauchy formula for homological invariance. Examples.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic. Examples. Laurent series.
Series expansion of a holomorphic function in a circular crown in the Laurent series. Isolated singularities and the field of meromorphic functions. Examples. Statement of the classification theorem of isolated singularities and residual theorem: local and global versions.
Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Ω)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Ω)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Analisi funzionale, H. Brézis, Liguori Editore

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Basic fluid modelling: conservation laws, viscid and inviscid models, incompressibility constraint. Approximate formulations (Euler, Stokes, Shallow Water Equations). Classical, weak and entropic solutions.
Finite difference numerical methods for Computational Fluid Dynamics: conservative schemes, Vorticity-Streamfunction methods, projection methods.

Note: The contents of the course have changed. For this reason, the details of the syllabus will be decided during the course itself.

R. J. LeVeque, Numerical methods for conservation laws, Birkhauser

L. Quartapelle, Numerical solution of the incompressible Navier-Stokes Equations, Springer

Additional material provided by the teacher.

Classification of the semilinear partial differential equations in any dimension. Classification in 2 dimensions and canonical forms. Wave equation in an interval using the separation of variables method: homogeneous case, general case. Wave equation on the line: D’Alambert’s solution. The half line case. Wave equation in R3.
Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc- tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa- tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.

A.N. Tichonov, A.A. Samarskij Equazioni della Fisica Matematica Edizioni MIR

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense
D.M. Burton, Elementary Number theory, McGraw-Hill, (sesta edizione 2007)
G.A. Jones e J.M. Jones, Elementary Number theory, Springer, 1998 (ristampa 2008)
H. Davenport, Aritmetica superiore. Una introduzione alla teoria dei numeri, Zanichelli, (1994)
G.H. Hardy e E.M. Wright, An introduction to the theory of numbers, Oxford University Press, (1979)
K.H. Rosen, Elementary number theory and its applications, Addison-Wesley (2000)

Sheaf theory and its use in on schemes

Preseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks
and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products.
Algebraic sheaves on a scheme. Quasi-coherent sheaves and coherent sheaves.

Cohomology of sheaves

Homological algebra in the category of modules over a ring. Flasque sheaves.
The cohomology of the sheaves using canonical resolution with flasque sheaves.

Cohomology of quasi-coherent and coherent sheaves on a scheme.

Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.

Invertible sheaves and linear systems

Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems. Base points. Linear systems, ample
and very ample sheaves. Amplitude criterion.

Notes from Prof. Sernesi
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Fermat's last Theorem for regular primes
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Basic fluid modelling: conservation laws, viscid and inviscid models, incompressibility constraint. Approximate formulations (Euler, Stokes, Shallow Water Equations). Classical, weak and entropic solutions.
Finite difference numerical methods for Computational Fluid Dynamics: conservative schemes, Vorticity-Streamfunction methods, projection methods.

Note: The contents of the course have changed. For this reason, the details of the syllabus will be decided during the course itself.

R. J. LeVeque, Numerical methods for conservation laws, Birkhauser

L. Quartapelle, Numerical solution of the incompressible Navier-Stokes Equations, Springer

Additional material provided by the teacher.

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

A) STRUCTURAL RULES EXPRESSED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC, LUDICS
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

On line lecture notes

Definizione e prime proprietà delle curve ellittiche: richiami sulle curve algebriche piane, cubiche lisce, legge di gruppo. Invrainte j. Anello degli endorfismi di una curva ellittica: la somma e la composizione di isogenie è un'isogenia, l'annelo degli endomorfismi ha caratteristica zero. Curve ellittiche su un anello e algoritmo di fattorizzazione di Lenstra. Punti di torsione, curve ellittiche ordinarie e supersingolari. Morfismo di Frobenius, polinomio minimo del morfismo di Frobenius. Forma quadratica sull'anello degli endomorfismi, teorema di Hasse. Accoppiamento di Weil. Applicazioni delle curve ellittiche alla crittografia: scambio delle chiavi di Diffie-Helman, attaco MOV, backdoor nel genaratore di numeri primi basato sulle curve ellittiche. Cenni alla crittografia basate sulle isogenie (in particolare su SIDH), formula di Vélu.

J. H. Silverman: The Arithmetic of Elliptic Curves, Graduate Studies in Mathematics.

L. C. Washington: Elliptic curves: Number Theory and Criptography, Chapman & Hall (CRC), second edition 2008.

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Introductory notions
Recap of Special Relativity. Lorentz transformations in Minkowski’s space. Vectors in Minkowski’s space. Basis of the tangent space.
Cotangents space and dual vectors in Minkowski’s space. Basis of cotangent space. Lorentz transformations of vectors and dual vectors. Tensors in Minkowski’s space. Properties of vectors, dual vectors and tensors in Minkowski’s space. Definition of symmetric and antisymmetric tensor. Symmetrization and antisymmetrization of a generic tensor. Metric in Minkowski’s space: definition and properties. Operations related to the metric: scalar products, rising and lowering indices of a tensor, contractions and trace of a tensor. Equivalence between inertial and gravitational mass. Weak Equivalence Principle (WEP), Einstein’s equivalence Principle (EEP).

Basic notions of differential geometry
Introduction to the notion of manifold. Definition and properties of maps. Injective and suriective maps (some examples included). Composition of charts. Invertible charts. Definition of diffeomorphism.
Definition of chart (or coordinate system). Definition of atlas. Definition of manifold. Product of manifolds.
Formal coordinate independent definition of vector. Demonstration that the dimension of the tangent space coincides with the one of the corresponding manifold. Basis (or coordinate system) of the tangent space. Coordinate transformations. Coordinate transformations of the components of a vector. Definition and properties of the tangent field. Definition of one parameter group of diffeomorphisms. Definition of integral curves. Commutator of two vectors. Coordinate independent definition of dual vector (one-form). Cotangent space and corresponding basis. Coordinate transformation of the components of a one-form. Coordinate independent definition of tensor. Demonstration that the partial derivative of a tensor is not a tensor.
Metric: signature and canonical form.
Tensor densities.
Differential forms. Wedge product. Exterior derivative. Closed and exact form. Poincarre Lemma (statement only). Hodge duality. Maxwell equations expressed in term of exterior derivative and hodge duality (only small reference).
Integration over a manifold: volume element in terms of the determinant of the metric.
Maps between manifold: pullback and pushforward. Pullback and pushforward associated to diffeomorphisms. Equivalence between diffeomorphisms and coordinate transformations. Vector field associated to diffeomorphisms. Lie Derivative: definition and general properties. Action of Lie’s derivative on scalars, vectors, one-forms and tensors. General Relativity as diffeomorphism invariant theory. Analogy between gauge transformations and diffeomorphisms.

Symmetries.
Notion of submanifold. Immersed and embedded submanifolds. Notion of hypersurface and boundary of a manifold.
Integration on manifolds again: differential form as generic volume element. Orientation and orientable manifold. Covering of the manifold through partition of unity. Integration of p-forms over submanifold. Demonstrations that the volume element can be expressed in terms of the determinant of the metric. Stokes theorem (no demonstration).


Connection, Covariant Derivative, Curvature
Lie’s Algebra and Lie’s group. Action from the right and from the left. Left- and right-invariant vectors. Structure constants. Examples of Lie groups. Maurer-Cartan forms. Maurer-Cartan’s equations. Action of Lie Groups on manifolds. Definition of free, effective and transitive action. Orbit and stabilizer.
Algebric definition of connection and covariant derivative. General properties of covariant derivatives. Action of coordinate transformations on the connection.
Demonstration that the difference of Christoffel coefficients associated to two different connections transforms as a tensor; torsion tensor, torsion-free and metric connection. Demonstrations that for any given metric exists a connection (metric connections) for which the covariant derivative of the metric is zero.
Formal construction of the covariant derivative from the notion of parallel transport (qualitative introduction).
Fiber bundle. Trivial and locally trivializable bundles. Local trivilizations. Maps between fiber bundles (notions).
Defintion of bundle atlas, G-atlas, G-structure. Fiber Bundle with structure group G. Definition of Principal Bundle. Definition of section of a bundle. Vector bundle and bundle of basis, definition and general properties. Relation between principle bundle, vectorbundle and bundle of frames (definition of associated vector bundle to a principal bundle.
Construction of the covariante derivative on a vectorbundle (only the knowledge of the fundamental logical steps is required for the exam).
Curvature tensor as 2-form on a fiber bundle. Geometrical interpretation of the curvature. Bianchi identity. Fiber metric. Ortogonal basis.
Connections and gauge theories: electromagnetism as simple example.
Soldering form. Choice of the gauge. Ortonormal and metric gauge.
Levi-Civita connection; Riemann’s tensor : definition and properties. Ricci’s tensor and scalar, Weyl’s tensor. Globally and locally inertial coordinates.

Einstein’s theory of gravity

Minimal coupling.
Particle in a gravitational field: affine parameter, self-parallel curves. Geodesic’s equations. Geodesic deviation.
Derivation of the Einstein’s equations from Newton’s limit.
Lagrangian derivations of Einstein’s equations.
General considerations on the structure of Einstein’s equations. Choice of the gauge. Energy conditions.
Symmetries and Killing vectors: version of Noether’s theorem from general relativity. Maximal number of linearly independent Killing vectors on a manifold. Homogenous and isotropic manifold. Spaces at constant curvature. Metric in spaces at constant curvature.

Notable solutions of Einstein’s equations

Static spherically symmetric spacetimes. Determination of Schwarzschild’s metric.
Cosmological solution. Spatially homogeneous and isotropic spacetime. Frieman’s Robertson-Walker metric. Friedman’s equations.
Coordinate singularities. Case of study: Schwarzschild radius. Rindler metric. Kruskal coordinates. Black hole solution.
Perturbation around a background metric. Case of study:perturbation of flat metric. Degrees of freedom. Linearized Einstein’s equations. Choice of the gauge. Linearized Einstein’s equations in vacuum: gravitational waves. Solutions in presence of the source (only few words).


Advanced concepts
Conformal transformations. Cotton’s tensor. Conformally flat metric. Demonstration of the theorem: a metric is conformally flat if and only if Weyl (Cotton) tensor is null. Conformal group. Conformal Killing vectors.
Alternative theories of gravity. Scalar-tensor theories. Jordan and Einstein’s frames.

1. S. Carrol Space time and Geometry: An Introduction to General Relativity (Addison Wesley, 2004);
2. R. Wald General Relativity (The Chicago Press, 1984);
3. B. Schutz A First Course in General Relativity (Cambridge Press)
4. people.sissa.it/~percacci/lectures/general/index.html
5. B. Schutz Geometrical Methods of Mathematical Physics (Cambridge Press)
6. S. Weinberg Gravitation and Cosmology-principles and application of the general theory of relativity (John Weiley & Sons, 1972);

Principles of Object Oriented Design
Abstraction, Polimorphism, Inheritance, Aggregation
Object Oriented Programming Models and UML
UML Use Case, Sequence, Class e Object, Deployment diagrams
Software Analysis and Developmenmt for Java Virtual Machine: I/O, Stream, Networking, Exception Handling
(Scientific, Real-time,...) Efficient Distributed Computing, Multithreading and Concurrency in Cloud e Mobile

Manuale di Java 9 De Sio Cesari Claudio Hoepli Informatica

CONTENTS OF THE LECTURES: the numbers in round brackets refer to the chapter and section of the textbook adopted.
Kinetic theory of gases. Boltzmann equation and H theorem. (1, Par.2.1,2.2,2.3,2.4)
Maxwell-Boltzmann distribution. (1, Par. 2.5)
Phase space and Liouville theorem. (1, Par. 3.1,3.2)
Gibbs ensembles. Micro canonical ensemble. Definition of entropy. (1, Par. 3.3,3.4)
The ideal gas in the micro canonical ensemble. (1, Par. 3.6)
The equipartition theorem. (1, Par. 3.5)
The canonical ensemble. (1, Par.4.1).
The partition function and the free energy. Fluctuations of energy in the canonical ensemble. (1 Par. 4.4)
The grand canonical ensemble. The grand potential. The ideal gas in the grand canonical ensemble. (1 Par. 4.3).
Fluctuations of the particle number. (1 Par. 4.4)
Classical theory of the linear response and fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein quantum statistics. ( 1, Par. 7.1)
The Fermi gas. The Sommerfeld expansion and the electron specific heat. (1, Par. 7.2)
The Bose gas. The Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)


Web page with additional material about the lecture course

https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi

Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.

Further reading:
2) K. Huang, Meccanica Statistica, Zanichelli, 1997.
3) L. Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri, 2003.
4) Joel L. Lebowitz, Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics, 71, S346 (1999).

Further information is available on
https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269

Taylor, J. - Introduzione all'analisi degli errori : lo studio delle incertezze nelle misure fisiche - Zanichelli Disponibile nella biblioteca Scientifica di Roma Tre

Dubi, A. - Monte Carlo applications in systems engineering - Wiley Disponibile nella biblioteca Scientifica di Roma Tre

Brief introduction to Julia, novel language for scientific computing. Introduction to geometric modeling and scientific visualization. Simplicial and cellular complexes. Chain complexes. Boundary and coboundary operators. Algebraic operators for incidence and adjacency. Duality of cell complexes. Extraction of geometric models from 3D images. Delaunay triangulations and Voronoi complexes. Alpha-shapes and alpha-complexes. Persistent homology. development of a collaborative progect: parallel LAR.

Lecture slides on Github:

Herbert Edelsbrunner and John Harer, Computational Topology. An Introduction, AMS, 2011.

Introduction to Machine Learning: what is machine learning; what is it aimed at, what are the problems; what are the theoretical tools used; overview of the topics that will be covered during the course.

Supervised and unsupervised learning; Model representation; The cost function; The gradient descent algorithm;

Linear regression; The gradient descent for linear regression; Logistic regression; The gradient descent for logistic regression; The normal equation;

The problem of classification; The representation of the hypothesis; The cost function; The one-vs-all method; The problem of overfitting; Regularization in linear and logistic regression;

The perceptron; Le Neural Networks; The Error-back propagation algorithm; Random initialization of weights;
Model selection; The train, validation and test set; Diagnosis by bias and variance; The learning curves; Error analysis;

Support Vector Machines;

K-means clustering;

Principal Components Analysis for dimensionality reduction;

Anomaly Detection algorithms;

Recommender Systems;

Large scale machine learning systems including parallelized and map-reduce systems;

C.M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006
R.O. Duda, P.E. Hart, D.G. Stork. Pattern Classification (2001) John Wiley & Sons.

The course aims to introduce students to the teaching of mathematics in first and second grade secondary schools, through a historical-epistemological approach to the basic concepts of elementary mathematics (arithmetic, geometry, algebra, probability, functions). In particular: the teaching of mathematics and its evolution; numerical systems; Euclid's axioms and postulates; non-Euclidean and locally Euclidean geometries; geometric constructions with ruler and compass and mathematical machines; elements of history of infinitesimal calculus. Outline of national indications.

GIORGIO ISRAEL, ANA MILLÁN GASCA, Pensare in matematica, Zanichelli, 2012.
ANA MILLÁN GASCA, All'inizio fu lo scriba, Mimesis, 2004
ENRICO GIUSTI, Analisi matematica 1, Bollati Boringhieri, 2002

TEACHING MATHEMATICS WITH THE HELP OF A COMPUTER: GEOGEBRA AND MATHEMATICA SOFTWARES.
COMMANDS FOR NUMERICAL AND SYMBOLIC CALCULUS, GRAPHICS VISUALIZATION, PARAMETRIC SURFACES AND CURVES WITH ANIMATIONS IN CHANGING PARAMETERS.
SOLVING PROBLEMS: TRIANGLE'S PROPERTIES IN EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY WITH EXAMPLES, APPROXIMATION OF PI AND OTHER IRRATIONAL NUMBERS, SOLUTIONS OF EQUATIONS AND INEQUALITIES,SYSTEMS OF EQUATIONS, DEFINING AND VISUALIZING GEOMETRICAL LOCI, FUNCTION INTEGRAL AND DERIVATIVES, APPROXIMATION OF SURFACE AREA.

LIST OF PROBLEMS GIVEN IN CLASS WITH VISUALIZATION AND SOLUTIONS WITH THE HELP OF SOFTWARE MATHEMATICA OR GEOGEBRA.

RENZO CADDEO, ALFRED GRAY LEZIONI DI GEOMETRIA DIFFERENZIALE - CURVE E SUPERFICI VOL. 1,
ED. CUEC (COOPERATIVA UNIVERSITARIA EDITRICE CAGLIARITANA)

The program includes two intertwined courses: themes that have a didactic interest and more specifically computational applicative themes. Classical topics (Euclidean geometry, point and line configurations ..) are chosen for their fallout in computer graphics, arguments of computational geometry are motivated by mathematical problems that have an elementary representation (Systems of polynomial equations in n unknowns ..).
 Based on the interests and requests of attending students, changes to parts of the program are possible.

 
Euclidean geometry: axioms, remarkable points in triangles, nine points circle, Morley's theorem, other theorems on triangles.
Affine geometry and barycentric coordinates, Ceva theorem, Menelaus theorem.
Projective geometry: axioms, the case of the plane over the finite field F2, Pappus and Desargues theorems, collineations and correlations.
Ordered geometry and the Sylvester problem on point collineation, generalizations.
Delaunay triangulations and Voronoi tessellation: properties and algorithms.
Ideals of polynomials, orderings of monomials and divisions between polynomials in several variables, Groebner bases.
Solving polynomial equations by elimination, by eigenvectors and eigenvalues, by resulting.
Polytopic geometry, mixed volume, Bernstein's theorem.

Materials, discussions, forum, videos on moodle https://matematicafisica.el.uniroma3.it

1) H.S.M. Coxeter Introduction to geometry, Wiley 1970;
2) D. Cox, J. Little, D. O’Shea Using Algebraic Geometry, GTM 185 Springer.
moreover
3) D. Cox, J. Little, D. O’Shea Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra UTM Springer
4) M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
5) S. Rebay, Tecniche di Generazione di Griglia per il Calcolo Scientifico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
6) B. Sturmfels, Polynomial equations and convex polytopes, American Mathematical Monthly 105 (1998) 907-922.
7) Shuhong Gao, Absolute Irreducibility of Polynomials via Newton Polytopes, J. of Algebra 237 (2001), 501-520.

The materials of the Earth: minerals, the lithogenetic processes, the lithogenetic cycle, the magmatic rocks, the sedimentary rocks, the metamorphic rocks, the bedding and the deformation of the rocks.
Volcanic phenomena: magma and volcanic activity, the main types of eruptions, shape of volcanic buildings, products of volcanic activity, the geographic distribution of volcanoes, volcanoes and man (the volcanic risk).
Seismic phenomena: the theory of elastic rebound, the seismic cycle, types of seismic waves and their propagation and registration, the force of an earthquake (scales of intensity and magnitude), the geographic distribution of earthquakes, the seismic activity and the man (seismic risk)
Plate tectonics: the internal structure of the Earth, the structure of the crust, the Earth's magnetic field, Earth’s internal heat, the convective mantle, from the hypothesis of the drift of the continents to the formulation of the theory of plate tectonics.
The Earth as an integrated system: interaction between the different systems of the planet (biosphere, atmosphere, hydrosphere, lithosphere, cryosphere), the earth's atmosphere, climate and meteorological phenomena, renewable and non-renewable natural resources.

Capire la Terra
J.P. Grotzinger, T-H Jordan
(Terza edizione italiana condotta sulla settima edizione americana)

Il Globo Terrestre e la sua evoluzione
E. L. Palmieri e M. Parotto
Sesta Edizione (2008)

Educational material distributed during the course

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Ω)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Ω)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Analisi funzionale, H. Brézis, Liguori Editore

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

Brownian motion (part I). Definition, properties and explicit construction of the Brownian motion. Markov property. Strong Markov property and reflection principle.
Brownian motion (part II). Multidimensional Brownian motion. Harmonic function and Dirichlet problem. Solution of Dirichlet problem via Brownian motion (on smooth domains). Poisson problem and its solution on smooth domains. Law of the iterated logarithm. Skorohod embedding. Donsker's invariance principle with applications.
Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Ito's formula with applications. Multidimensional Ito's formula, general formula for stochastic integral.
Stochastic differential equations (SDE). Linear SDE with solutions. Theorem for existence and uniqueness of solutions of SDE. Partial differential equations. Feynman-Kac formula. Applications to financial mathematics (introduction to the Black-Scholes model).

- Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf
- An introduction to Stochastic Differential Equations (Evans)
- Brownian Motion and Stochastic Calculus (Karatzas and Shreve, 1998) https://www.springer.com/gp/book/9780387976556
- An Introduction to Stochastic Calculus with Applications to Finance (Ovidiu Calin)
https://people.emich.edu/ocalin/Teaching_files/D18N.pdf

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

Refer to the web-page of the course

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples

Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

The program focuses on major themes of Biological Evolution of animal and plant organisms.
Light and Life, Evolution of photomorphogenesis, Secondary metabolites of plants, The conquest of emerged lands, plant-insect co-evolution.

There are no specific textbooks. Students can refer to the material shown in the classroom and made available.
Recommended readings:

Taiz L., Zeiger E. - Fisiologia Vegetale - Piccin Editore

Rascio N. : Elementi di Fisiologia vegetale - EdiSES

PLANT PHYSIOLOGY (IN ENGLISH LANGUAGE)
LINCOLN TAIZ- EDUARDO ZEIGER
SINAUER ASSOCIATES (FIFTH EDITION)

Zocchi A.;Imparare ad Imparare; Amazon Kindle eBook/cartaceo copertina flessibile


The teacher receives every day (Monday-Friday) by appointment via institutional email

1. ATOMIC THEORY AND ATOMIC STRUCTURE. Atoms, molecules, moles; atomic and molecular weight. Atomic models: Rutheford, Bohr. Quantum theory, quantum numbers and energy levels. Polyelectronic atoms; periodic system.
2. CHEMICAL BONDS. Ionic bond. Covalent bond: and bonds. Polyatomic molecules: molecular structure. Hybridization and resonance. Molecular orbital. Metallic bond. Intermolecular forces.
3. NOMENCLATURE AND CHEMICAL REACTIONS. Oxides, hydroxides, acids, salts, ions. Balancing chemical reactions: redox reactions.
4. STATES OF AGGREGATION. Gas state, ideal gas law. Solid state: ionic, covalent, molecular and metallic solids. Conductors, semiconductors, insulators. Liquid and amorphous states. Phase transitions and phase diagrams.
5. SOLUTIONS. Concentration, colligative properties; electrolyte solutions.
6. THERMODYNAMICS. Matter, energy, heat, first and second principles; enthalpy, entropy, free energy.
7. CHEMICAL EQUILIBRIUM. Equilibrium constant and free energy. Gas-phase and heterogeneous equilibria. Le Chatelier’s principle. Van’t Hoff equation.
8. EQUILIBRIA IN SOLUTION. Acid-base equilibria: acids and bases, pH, dissociation constant, polyprotic acids, hydrolysis, buffers. Acid-base titrations and pH indicators. Solubility equilibria: solubility product, common ion effect.
9. ELECTROCHEMISTRY. Batteries, electrode potentials, Nernst’s equation. Electrolysis.
10. CHEMICAL KINETICS. Reaction speed, speed constant. Influence of the temperature on the reaction speed: Arrhenius equation. Catalysts.
Numerical exercises on all the listed subjects.

P.W.Atkins, L. Jones; CHEMISTRY: MOLECULES, MATTER, AND CHANGE

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

Part 1. Affine algebraic geometry
Algebraic preliminaries: Noetherian rings and Hilbert basis theorem.
Zariski topology
Plane curves.
Irreducible sets and e prime ideals. Decomposition into irreducible components.
Nullstellensatz with proof.
Radical ideals and Hilbert correspondence with closed subsets. Algebra of a closed set.
Regular maps and isomorphisms. Corrispondence between algebras and closed sets.
Dimension.
Characterization of dominant maps.
Principal open substes.
Products and Projections.

Part 2. Projective algebraic geometry.
Projective space: Zariski topology , correspondence between homogeneous radical ideals and projective closed sets. Projective hypersurfaces and their spaces. Dual projective space.
Quasiprojective varieties. Rational functions and maps on quasiprojective varieties.
Algebras of rational and regular functions on quasiprojective varieties.
Affine and r projective varieties . Quasiprojective varietie as finite union of affine varieties.
Dimension.
Birational equivalence.
Automorphisms and e projectivities of projective spaces.
Rational normal curves
Veronese maps and e varieties.
Products, Segre maps and varieties.
Projections from closed subsets.
Resultant and conservation of projective closure.
Blowing-up of projective space in a point.
Graoph of a rational map and resolution of rational maps.


Part 3: Local geometry
Local ring of a variety at a point.
Cotangentspace and zariski tangent space..
Singolari and nonsingolar points.
Upper-semicontinuity of the dimension of the fibers of a morphism.
Locally factorial varieties and their properties.
Bertini's theorem on nonsingularity of fibers.
Geometriy of curves. Examples of nonrational nonsingular curve.
Normal varieties and e normalization. Esistence and e uniqueness of normalization of affine varieties.

Part 4 Divisors
Divisors and divisor group. Principal divisors.
Linear equivalence on divisors.
Class group (Picard group)
Linear systems of divisors associated to projective varieties.
Linear system of a general divisor.

(1) I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
(2) J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

Short introduction to Julia for scientific programming. Introduction to parallel aechitectures. Designi principles of parallel algorithms. Prallel and distributed programming with Julia. Comunication and sincronization primitives: MPI paradigm. Directive-based languages: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: mentions to BLAS, LAPACK, scaLAPACK. Sparse linear systems. Mentions to CombBLAS and GraphBLAS.

1. [Lecture slides and diary](https://github.com/cvdlab-courses/pdc/blob/master/schedule.md)

2. [Learning Julia](https://www.manning.com/books/julia-in-action)

3. Blaise N. Barney, [HPC Training Materials](https://computing.llnl.gov/tutorials/parallel_comp/), per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

4. J. Dongarra, J. Kurzak, J. Demmel, M. Heroux, [Linear Algebra Libraries for High- Performance Computing: Scientific Computing with Multicore and Accelerators](http://www.netlib.org/utk/people/JackDongarra/SLIDES/sc2011-tutorial.pdf), SuperComputing 2011 (SC11)

Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Classification of the semilinear partial differential equations in any dimension. Classification in 2 dimensions and canonical forms. Wave equation in an interval using the separation of variables method: homogeneous case, general case. Wave equation on the line: D’Alambert’s solution. The half line case. Wave equation in R3.
Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc- tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa- tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.

A.N. Tichonov, A.A. Samarskij Equazioni della Fisica Matematica Edizioni MIR

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them (for example, because it gives a more efficient algorithm, or because the solution is becomes more resilient against and adversary, or simply because it gives a simple and elegant solution to an apparently complicated problem).

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Allocation process like balls into bins and random data structures via hash functions
* Branching processes and spread of infections
* Probabilistic method and its application to combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data