Optional group:
CURRICULUM TEORICO SCEGLIERE QUATTRO INSEGNAMENTI (30 CFU) NEI SEGUENTI SSD MAT/01,02,03,05 TRA LE ATTIVITÀ CARATTERIZZANTI (B) - (show)
 |
30
|
|
|
|
|
|
|
|
|
20410408 -
AL310 - ELEMENTS OF ADVANCED ALGEBRA
(objectives)
Acquire a good knowledge of the concepts and methods of the theory of polynomial equations in one variable. Learn how to apply the techniques and methods of abstract algebra. Understand and apply the fundamental theorem of Galois correspondence to study the "complexity" of a polynomial.
-
Derived from
20410408 AL310 - ISTITUZIONI DI ALGEBRA SUPERIORE in Matematica L-35 PAPPALARDI FRANCESCO, TOLLI FILIPPO
( syllabus)
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.
( reference books)
J. S. Milne,Fields and Galois Theory.Course Notes, (2015).
|
9
|
MAT/02
|
48
|
24
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410445 -
AL410 - COMMUTATIVE ALGEBRA
(objectives)
Acquire a good knowledge of some methods and fundamental results in the study of the commutative rings and their modules, with particular reference to the study of ring classes of interest for the algebraic theory of numbers and for algebraic geometry.
-
TARTARONE FRANCESCA
( syllabus)
1. Modules
Modules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.
( reference books)
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
|
9
|
MAT/02
|
48
|
24
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410451 -
LM410 -THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.
|
|
|
20410455 -
LM420 - THEOREMS IN LOGIC 2
(objectives)
To support the students into an in-depth analysis of the main results of first order classical logic and to study some of their remarkable consequences.
-
Derived from
20710122 TEOREMI SULLA LOGICA, 2 in Scienze filosofiche LM-78 TORTORA DE FALCO LORENZO
( syllabus)
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.
( reference books)
V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018
|
6
|
MAT/01
|
36
|
-
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410425 -
GE460- GRAPH THEORY
(objectives)
Provide tools and methods for graph theory.
-
Derived from
20410425 GE460 - TEORIA DEI GRAFI in Scienze Computazionali LM-40 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
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.
( reference books)
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.
|
6
|
MAT/03
|
48
|
12
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410518 -
AM420 - SOBOLEV SPACES AND PARTIAL DERIVATIVE EQUATIONS
(objectives)
To acquire a good knowledge of the general methods andÿclassical techniques necessary for the study ofÿweak solutions of partial differential equations.
-
HAUS EMANUELE
( syllabus)
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
( reference books)
Functional analysis, H. Bre'zis
-
FEOLA ROBERTO
( syllabus)
Definition and elementary properties of the Sobolev spaces. Extensions theorem. Sobolev inequalities. Trace operator, compactness. Duality. Fourier transform method. Second order elliptic equations: existence of weak solutions. Regularity:interior/boundary. Maximum principles. Arguments of evolutions problems: the wave equation.
( reference books)
Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
Lawrence C. Evans - Partial Differential Equations. AMS
|
6
|
MAT/05
|
48
|
12
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410627 -
TN410 - INTRODUCTION TO NUMBER THEORY
(objectives)
Acquire a good knowledge of the concepts and methods of the elementary number theory, with particular reference to the study of the Diophantine equations and congruence equations. Provide prerequisites for more advanced courses of algebraic and analytical number theory.
-
BARROERO FABRIZIO
( syllabus)
Arithmetic functions and their properties:
-Definition and Dirichlet convolution.
-Number and sum of divisors function.
-Möbius function.
-Euler function.
Congruences:
-Sets of residues.
-Polynomial congruences.
-Primitive roots.
Quadratic residues:
-Legendre symbol.
-Quadratic reciprocity.
-Jacobi symbol.
Sums of squares:
-Sums of two squares.
-Number of representations.
-Sums of four squares.
-Sums of three squares.
Continued fractions and diophantine approximation:
-Simple continued fractions.
-Continued fractions and diophantine approximation.
-Infinite simple continued fractions.
-Periodic continued fractions.
-Pell's equation.
-Liouville's Theorem.
( reference books)
Lecture notes
Note di W. Chen
http://www.williamchen-mathematics.info/lnentfolder/lnent.html
An Introduction to the Theory of Numbers by G. H. Hardy, E. M. Wright
M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense
|
6
|
MAT/02
|
48
|
12
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410444 -
GE430 - RIEMANNIAN GEOMETRY
(objectives)
Introdue to the study of Riemannian geometry, in particular by addressing the theorems of Gauss-Bonnet and Hopf-Rinow.
-
SCHAFFLER LUCA
( syllabus)
We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.
( reference books)
Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.
|
6
|
MAT/03
|
48
|
12
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410407 -
AC310 - Complex analysis
(objectives)
To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.
|
9
|
MAT/03
|
48
|
24
|
-
|
-
|
Core compulsory activities
|
ITA |
|
20410637 -
AM450 - FUNCTIONAL ANALYSIS
(objectives)
To acquire a good knowledge of functional analysis: Banach and Hilbert spaces, weak topologies, linear and continuous operators, compact operators, spectral theory.
|
9
|
MAT/05
|
48
|
24
|
-
|
-
|
Core compulsory activities
|
ITA |
|
|
Optional group:
COMUNE AI 2 CURRICULA TEORICO E MODELLISTICO: SCEGLIERE QUATTRO INSEGNAMENTI (30 CFU) TRA LE ATTIVITÀ AFFINI INTEGRATIVE (C) - (show)
|
30
|
|
|
|
|
|
|
|
|
20410408 -
AL310 - ELEMENTS OF ADVANCED ALGEBRA
(objectives)
Acquire a good knowledge of the concepts and methods of the theory of polynomial equations in one variable. Learn how to apply the techniques and methods of abstract algebra. Understand and apply the fundamental theorem of Galois correspondence to study the "complexity" of a polynomial.
-
Derived from
20410408 AL310 - ISTITUZIONI DI ALGEBRA SUPERIORE in Matematica L-35 PAPPALARDI FRANCESCO, TOLLI FILIPPO
( syllabus)
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.
( reference books)
J. S. Milne,Fields and Galois Theory.Course Notes, (2015).
|
9
|
MAT/02
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410445 -
AL410 - COMMUTATIVE ALGEBRA
(objectives)
Acquire a good knowledge of some methods and fundamental results in the study of the commutative rings and their modules, with particular reference to the study of ring classes of interest for the algebraic theory of numbers and for algebraic geometry.
-
Derived from
20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 TARTARONE FRANCESCA
( syllabus)
1. Modules
Modules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.
4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.
( reference books)
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
|
9
|
MAT/02
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410416 -
FM410-Complements of Analytical Mechanics
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
|
|
|
20410416-1 -
FM410-Complements of Analytical Mechanics - MODULE A
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
-
Derived from
20410084 COMPLEMENTI DI MECCANICA ANALITICA - MOD A in Fisica L-30 GENTILE GUIDO
( syllabus)
Linear dynamic systems. Forced harmonic oscillation in the presence or absence of dissipation. Limit sets and limit cycles. Planar systems. Gradient systems. Stability theorems. Lotka-Volterra equations. Van der pol equation. Epidemiologic models (epidemic SIR, endemic SIR and endemic SEIR).
( reference books)
G. Gentile, Introduzione ai sistemi dinamici. 1. Equazioni differenziali ordinarie, analisi qualitativa e alcune applicazioni, available onlineG. Gentile, Introduzione ai sistemi dinamici. 2. Meccanica lagrangiana e hamiltoniana, available online
|
3
|
MAT/07
|
30
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410416-2 -
FM410-Complements of Analytical Mechanics - Module B
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
-
Derived from
20410085 COMPLEMENTI DI MECCANICA ANALITICA - MOD. B in Fisica L-30 GENTILE GUIDO
( syllabus)
Euler's angles. Euler's equation for the dynamics of the rigid body. Spinning top of Lagrange. Flow box theorem. Noether's theorem inm the case of more groups of symmetries. Theory of small oscillations.Perturbation theory. Homological equation. Isochronous and anisochronous systems. Birkhoff series. All-order perturbation theory for isochronous systems and Nekhoroshev's theorem.
( reference books)
G. Gentile, Introduzione ai sistemi dinamici. 1. Equazioni differenziali ordinarie, analisi qualitativa e alcune applicazioni, available onlineG. Gentile, Introduzione ai sistemi dinamici. 2. Meccanica lagrangiana e hamiltoniana, available online
|
3
|
MAT/07
|
30
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410451 -
LM410 -THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.
|
|
|
20410438 -
MF410 - Computational Finance
(objectives)
Basic knowledge of financial markets, introduction to computational and theoretical models for quantitative finance, portoflio optimization, risk analysis. The computational aspects are mostly developed within the Matlab environment.
-
Derived from
21201730 FINANZA COMPUTAZIONALE in Finanza e impresa LM-16 CESARONE FRANCESCO
( syllabus)
MODULE 1
1 A rapid introduction to MATLAB
1.1 MATLAB basics: Preliminary elements; Variable assignment; Workspace; Arithmetic operations; Vectors and matrices; Standard operations of linear algebra; Element-by-element multiplication and division; Colon (:) operator; Predefined function; inline Function; Anonymous Function.
1.2 M-file: Script and Function
1.3 Programming fundamentals: if, else, and elseif scheme; for loops; while loops
1.4 Matlab graphics
1.5 Preliminary exercises on programming
1.6 Exercises on the financial evaluation basics
MODULE 2
2 Preliminary elements on Probability Theory and Statistics
2.1 Random variables
2.2 Probability distributions
2.3 Continuous random variable
2.4 Higher-order moments and synthetic indices of a distribution
2.5 Some probability distributions: Uniform, Normal, Log-normal, Chi-square, Student-t
3 Linear and Non-linear Programming
3.1 Some Matlab built-in functions for optimization problems
3.2 Multi-objective optimization: Determining the efficient frontier
4 Portfolio Optimization
4.1 Portfolio of equities: Prices and returns
4.2 Risk-return analysis: Mean-Variance; Effects of the diversification in an Equally Weighted portfolio; Mean-MAD; Mean-MinMax; VaR; Mean-CVaR; Mean-Gini portfolios
4.3 Bond portfolio immunization
MODULE 3
5 Further elements on Probability Theory and Statistics
5.1 Introduction to the Monte Carlo simulation
5.2 Stochastic processes: Brownian motion; Ito’s Lemma; Geometrical Brownian motion
6 Pricing of derivatives with an underlying security
6.1 Binomial model (CRR): A replicating portfolio of stocks and bonds; Calibration of the model; Multi-period case
6.2 Black-Scholes model: Assumptions of the model; Pricing of a European call; Pricing equation for a call; Implied Volatility
6.3 Option Pricing with Monte Carlo Method: Solution in integral form; Path Dependent Derivatives
( reference books)
F Cesarone (2020), Computational Finance. MATLAB oriented modeling, Routledge-Giappichelli Studies in Business and Management, ISBN 978-0-367-49303-5
https://www.giappichelli.it/computational-finance
|
9
|
SECS-S/06
|
60
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410419 -
MS410-Statistical Mechanics
(objectives)
To acquire the mathematical basic techniques of statistical mechanics for interacting particle or spin systems, including the study of Gibbs measures and phase transition phenomena, and apply them to some concrete models, such as the Ising model in dimension d = 1,2 and in the mean field approximation.
-
Derived from
20410419 MS410-MECCANICA STATISTICA in Scienze Computazionali LM-40 GIULIANI ALESSANDRO
( syllabus)
INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states
THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1
( reference books)
S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html
|
9
|
MAT/07
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410420 -
AN420 - NUMERICAL ANALYSIS 2
(objectives)
Introduce to the study and implementation of more advanced numerical approximation techniques, in particular related to approximate solution of ordinary differential equations, and to a further advanced topic to be chosen between the optimization and the fundamentals of approximation of partial differential equations.
-
Derived from
20410420 AN420 - ANALISI NUMERICA 2 in Scienze Computazionali LM-40 CACACE SIMONE
( syllabus)
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)
( reference books)
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
|
9
|
MAT/08
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410442 -
IN420 - Information Theory
(objectives)
Introduce key questions in the theory of signal transmission and quantitative analysis of signals, such as the notions of entropy and mutual information. Show the underlying algebraic structure. Apply the fundamental concepts to code theory, data compression and cryptography.
-
Derived from
20410442 IN420 - TEORIA DELL'INFORMAZIONE in Scienze Computazionali LM-40 BONIFACI VINCENZO
( syllabus)
1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.
2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.
3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.
4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.
( reference books)
David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.
|
9
|
INF/01
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410455 -
LM420 - THEOREMS IN LOGIC 2
(objectives)
To support the students into an in-depth analysis of the main results of first order classical logic and to study some of their remarkable consequences.
-
Derived from
20710122 TEOREMI SULLA LOGICA, 2 in Scienze filosofiche LM-78 TORTORA DE FALCO LORENZO
( syllabus)
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.
( reference books)
V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018
|
6
|
MAT/01
|
36
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410459 -
MC430 - LABORATORY: DIDACTICS FOR MATHEMATICS
(objectives)
1. Mathematics software, with particular attention to their use for teaching mathematics in school. 2. Analysis of the potential and criticality of the use of technological tools for teaching and learning mathematics.
-
Derived from
20410459 MC430 - LABORATORIO DI DIDATTICA DELLA MATEMATICA in Matematica LM-40 FALCOLINI CORRADO
( syllabus)
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.
( reference books)
LIST OF PROBLEMS GIVEN IN CLASS WITH VISUALIZATION AND SOLUTIONS WITH THE HELP OF SOFTWARE MATHEMATICA OR GEOGEBRA.
|
6
|
MAT/04
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410435 -
FS440 - Data Acquisition and Experimental Control
(objectives)
The lectures and laboratories allow the student to learn the basic concepts pinpointing the data acquisition of a high energy physics experiment with specific regard to the data collection, control of the experiment and monitoring.
-
Derived from
20401070 ACQUISIZIONE DATI E CONTROLLO DI ESPERIMENTI in Fisica LM-17 N0 RUGGIERI FEDERICO
( syllabus)
The course aim is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics: -Introduction to DAQ-Parallelism and Pipelining systems -Derandomization-DAQ and Trigger-Data Transmission -Front End Electronics-Trigger-Architecture Computing Systems-Real Time Systems-Real Time Operating Systems -C Language-TCP / IP Network Protocols-DAQ-Architecture Building -VME Bus-Run Control-Farming-Data Archiving During the course, laboratory exercises will take place with the execution of simple examples of: - reading and data transfer systems through pipe mechanisms with concurrent processes; - signal-based trigger simulation programs; - Run Control program for activation and termination of processes; - configuration and reading of data from board on VME bus.
( reference books)
Lecture notes prepared by the teacher on the basis of the slides presented and available on the Moodle server:
https://matematicafisica.el.uniroma3.it
|
6
|
FIS/04
|
60
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410424 -
IN450 - ALGORITHMS FOR CRYPTOGRAPHY
(objectives)
Acquire the knowledge of the main encryption algorithms. Deepen the mathematical skills necessary for the description of the algorithms. Acquire the cryptanalysis techniques used in the assessment of the security level provided by the encryption systems.
-
Derived from
20410424 IN450- ALGORITMI PER LA CRITTOGRAFIA in Scienze Computazionali LM-40 PEDICINI MARCO
( syllabus)
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 cypher. Product cyphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine cyphers, for substitution cypher (frequency analysis), for Vigenere cypher: Kasiski test, coincidence index; cryptoanalysis of Hill's cypher and LFSR: algebraic attacks, cube attack.
2. Application of Shannon theory to cryptography
- Security of cyphers: 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 cyphers
- 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 cyphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the GCD and inverse; Operating modes for block cyphers.
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).
( reference books)
[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.
|
6
|
INF/01
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410425 -
GE460- GRAPH THEORY
(objectives)
Provide tools and methods for graph theory.
-
Derived from
20410425 GE460 - TEORIA DEI GRAFI in Scienze Computazionali LM-40 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
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.
( reference books)
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.
|
6
|
MAT/03
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410428 -
CR510 – ELLIPTIC CRYPTOSYSTEMS
(objectives)
Acquire a basic knowledge of the concepts and methods related to the theory of public key cryptography using the group of points of an elliptic curve on a finite field. Apply the theory of elliptic curves to classical problems of computational number theory such as factorization and primality testing.
|
6
|
MAT/02
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410470 -
FM510 - MATHEMATICAL PHYSICS APPLICATIONS
(objectives)
To apply methods and tools of mathematical physics to some classes of models of dynamical systems and statistical mechanics, through both theoretical lectures and numerous practical exercises carried out in the computer lab.
-
Derived from
20410470 FM510 - APPLICAZIONI DELLA FISICA MATEMATICA in Scienze Computazionali LM-40 TERESI LUCIANO, SCOPPOLA ELISABETTA, D'AUTILIA ROBERTO
( syllabus)
Dynamical system and Reaction Diffusion problems; morphogenesis.
( reference books)
Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018
https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0
A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952
|
9
|
MAT/07
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410462 -
GE510 - ALGEBRAIC GEOMETRY 2
(objectives)
Introduce to the study of algebraic geometry, with particular emphasis on beams, schemes and cohomology.
-
LOPEZ ANGELO
( syllabus)
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.
( reference books)
Notes from Prof. Lopez, 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).
|
6
|
MAT/03
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410434 -
FS450 - Elements of Statistical Mechanics
(objectives)
Gain knowledge of fundamental principles of statistical mechanics for classical and quantum systems.
|
6
|
FIS/02
|
60
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410518 -
AM420 - SOBOLEV SPACES AND PARTIAL DERIVATIVE EQUATIONS
(objectives)
To acquire a good knowledge of the general methods andÿclassical techniques necessary for the study ofÿweak solutions of partial differential equations.
-
Derived from
20410518 AM420 - SPAZI DI SOBOLEV ED EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM-40 HAUS EMANUELE, FEOLA ROBERTO
( syllabus)
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
( reference books)
Functional analysis, H. Bre'zis
|
6
|
MAT/05
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410557 -
GE530-Linear algebra for Machine Learning
(objectives)
Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization–and above all a full explanation of deep learning.
-
Derived from
20410557 GE530 - ALGEBRA LINEARE PER IL MACHINE LEARNING in Scienze Computazionali LM-40 TERESI LUCIANO, FERMI DAVIDE
( syllabus)
Highlights of Linear Algebra:
Matrix-matrix multiplication; column & row space; rank
The four fundamental subspaces of linear algebra
Fundamentals of Matrix factorizations:
A=LU rows & columns point of view
A=LU elimination & factorization; permutations
A=RU=VU; Orthogonal matrices
Eigensystems and Linear ODE
Intro to PSym; the energy function
Gradient and Hessian
Singular Value Decomposition
Eckart-Young; derivative of a matrix norm
Principal Component Analysis
Generalized evectors;
Norms
Least Squares
Convexity & Newton’s method
Newton & L-M method; Recap of non-linear regression
Lagrange multipliers
Machine Learning:
Gradient Descend; exact line search; GD in action; GD with Matlab
Learning & Loss; Intro to Deep Neural Network; DNN with Matlab
Loss functions: Quadratic VS Cross entropy
Stocastics Gradient Descend (SGD) & Kaczmarcz; SGD convergence rates & ADAM
Matlab interface for DNN
Construction of DNN: the key steps
Backpropagation and the Chain Rule
Machine Learning examples with Wolfram Mathematica
Convolutional NN + Mathematica examples of 1D convolution
Convolution and 2D filters + Mathematica examples of 2D convolution
Matlab Live Script, Network Designer, Pretrained Net
( reference books)
G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press
M. Nielsen,
Neural Networks and Deep Learning (free online book)
http://neuralnetworksanddeeplearning.com
Various authors,
Distill, dedicated to clear explanations of machine learning
https://distill.pub
|
9
|
MAT/03
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410560 -
IN400- Python and MATLAB programming
|
|
|
20410566 -
FS470 - Principles of astrophysics
(objectives)
Provide the student with a first view of some of the fundamental topics of Astrophysics and Cosmology using the mathematical and physical knowledge acquired in the first two years
-
Derived from
20410499 Principi di Astrofisica in Fisica L-30 LA FRANCA FABIO, MATT GIORGIO
( syllabus)
Topics Part A
• Coordinates and Telescopes
• Elements of Spectroscopy
• Stars and Stellar Evolution • Galaxies
• Active Galactic Nuclei
Program Part A
• Overview
• Celestial coordinates (1.3)
• Telescopes and resolving power (6.1)
• Parallax distance (3.1)
• Flux, brightness, apparent and absolute magnitudes, colors (3.2, 3.3, 3.6)
• The black body (3.4, 3.5)
• Hertzsprung-Russel diagram (8.2)
• Open and globular clusters: position, stellar populations and HR diagram (13.3)
• White dwarfs, Novae and SuperNovae (notes and partly in 15 and 16)
• The classification of galaxies (24.1)
• The rotation curve of galaxies and dark matter (25.3)
• The center of the Galaxy and its Black Hole (25.4)
• Hubble's law and expansion of the Universe (27.2)
• Probability of collision between stars and galaxies (handouts)
• Black Holes: outline of General Relativity (outline 17)
• Active Galactic Nuclei (28.1, 28.2, 28.3)
Topics Part B
• Structure and stellar evolution
• Elements of Spectroscopy
• Distances and expansion of the Universe • Galaxies
• GRB and gravitational waves
Program Part B
• Acretion disks and X-ray emission in Active Galactic Nuclei (28.2)
• Stars of Neutrons and Pulsars (16.6, 16.7)
• Gamma Ray Bursts (handouts)
• Gravitational Waves (lecture notes)
• Spectroscopy: eq. Boltzmann-excitation and Saha-ionization (8.1)
• Spectroscopy: speed, temperature and density measurements (handouts)
• Eq. of star structure, time and Kelvin-Helmholtz instability (11.1-4)
• Nuclear reactions of hydrogen (11.3)
• Jeans mass of gravitational collapse, free-fall time and Initial Mass Function (12.2, 12.3)
• The Milky Way and the local group (25.1, 25.2)
• Metallicity (25.2)
• Transit of Venus and measurement of the Earth-Sun distance (handouts)
• Distance scale (27.1)
• Hubble's law and expansion of the Universe (27.2)
• Local Group, Clusters of Galaxies, large scale structure of the Universe (27.3)
• The Big Bang and the background radiation (29.2 brief notes and lecture notes)
( reference books)
A copy of the lecture notes can be downloaded from the course website.
In brackets, the paragraphs from “An Introduction to Modern Astrophysics, II ed. - B.W. Carrol, D.A. Ostlie - Ed. Pearson, Addison Wesley ”(copies available in the library). The discussion in the course has been simplified compared to what is reported in the text. Alternative text in Italian: Attilio Ferrari, Stars, Galaxies, Universe - Fundamentals of Astrophysics - Ed. Springer
|
6
|
FIS/05
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410592 -
LM400 - INTRODUZIONE ALLA LOGICA
(objectives)
To Introduce students to themes, concepts, methods and results of logic that are at the basis of every discipline, in order to provide students - having any kind of background- with a deep interdisciplinary approach and an appropriate training for school teaching
-
ABRUSCI VITO MICHELE
( syllabus)
1. The themes of logic
2. Classical logic: propositions, proofs
3. Classical logic: connectives
4. Classical logic: types, variables, quantifiers
5. First-order Classical logic
6. Classes and sets
7. Codes, digits, Boolean algebra
8. Turing Machine
9. Axiomatization and formalization of first-order classical logic
10. Logic and other disciplines
( reference books)
V. Michele Abrusci, LOGICA - Lezioni di primo livello, Quarta edizione, Wolters Kluwer, 2018
|
6
|
M-FIL/02
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410529 -
LM510 - LOGICAL THEORIES 1
(objectives)
Address some questions of the theory of the proof of the twentieth century, in connection with the themes of contemporary research
-
Derived from
20710091 TEORIE LOGICHE 1 - LM in Scienze filosofiche LM-78 MAIELI ROBERTO
( syllabus)
A) STRUCTURAL RULES INTERPRETED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE NON-DETERMINISM: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC AND PROOF SEARCH
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
( reference books)
NOTES AND SLIDES AVAILABLE ON THE COURSE WEB PAGE
https://sites.google.com/view/lm510/
|
6
|
MAT/01
|
36
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410627 -
TN410 - INTRODUCTION TO NUMBER THEORY
(objectives)
Acquire a good knowledge of the concepts and methods of the elementary number theory, with particular reference to the study of the Diophantine equations and congruence equations. Provide prerequisites for more advanced courses of algebraic and analytical number theory.
-
Derived from
20410627 TN410 - INTRODUZIONE ALLA TEORIA DEI NUMERI in Matematica LM-40 BARROERO FABRIZIO
( syllabus)
Arithmetic functions and their properties:
-Definition and Dirichlet convolution.
-Number and sum of divisors function.
-Möbius function.
-Euler function.
Congruences:
-Sets of residues.
-Polynomial congruences.
-Primitive roots.
Quadratic residues:
-Legendre symbol.
-Quadratic reciprocity.
-Jacobi symbol.
Sums of squares:
-Sums of two squares.
-Number of representations.
-Sums of four squares.
-Sums of three squares.
Continued fractions and diophantine approximation:
-Simple continued fractions.
-Continued fractions and diophantine approximation.
-Infinite simple continued fractions.
-Periodic continued fractions.
-Pell's equation.
-Liouville's Theorem.
( reference books)
Lecture notes
Note di W. Chen
http://www.williamchen-mathematics.info/lnentfolder/lnent.html
An Introduction to the Theory of Numbers by G. H. Hardy, E. M. Wright
M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense
|
6
|
MAT/02
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410444 -
GE430 - RIEMANNIAN GEOMETRY
(objectives)
Introdue to the study of Riemannian geometry, in particular by addressing the theorems of Gauss-Bonnet and Hopf-Rinow.
-
Derived from
20410444 GE430 - GEOMETRIA RIEMANNIANA in Matematica LM-40 SCHAFFLER LUCA
( syllabus)
We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.
( reference books)
Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.
|
6
|
MAT/03
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410626 -
IN440 - COMBINATORIAL OPTIMISATION
(objectives)
Acquire skills on key solution techniques for combinatorial optimization problems; improve the skills on graph theory; acquire advanced technical skills for designing, analyzing and implementing algorithms aimed to solve optimization problems on graphs, trees and flow networks.
-
Derived from
20410626 IN440 - OTTIMIZZAZIONE COMBINATORIA in Scienze Computazionali LM-40 BONIFACI VINCENZO
( syllabus)
1. Optimization and combinatorial optimization problems. Enumeration of solutions.
2. Basics of algorithm analysis. Computational tractability. Asymptotic order of growth.
3. Graphs. Graph connectivity and graph traversal. Graph bipartiteness. Connectivity in directed graphs. Directed acyclic graphs and topological ordering.
4. Greedy algorithms. Interval scheduling. Optimal caching. Shortest paths in a graph. Minimum spanning trees.
5. Divide and conquer. Mergesort. Counting inversions. Closest pair of points.
6. Dynamic programming. Weighted interval scheduling. Principles of dynamic programming. Subset sums and knapsacks. All-pairs shortest paths. Shortest paths and distance vector protocols.
7. Network flow. Maximum flow and the Ford-Fulkerson algorithm. Maximum flows and minimum cuts in a network. Augmenting paths. Bipartite matching. Disjoint paths in directed and undirected graphs.
8. Computational intractability. Polynomial-time reductions. Reductions via "gadgets". Efficient certification and the definition of NP. NP-complete problems. Covering, packing, partitioning, sequencing, and numerical problems. Other examples.
( reference books)
Jon Kleinberg, Eva Tardos. Algorithm Design. Pearson Education, 2013.
|
9
|
MAT/09
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410407 -
AC310 - Complex analysis
(objectives)
To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.
|
9
|
MAT/03
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410571 -
FS520 – Complex networks
(objectives)
This course introduces students to the fascinating network science,
both from a theoretical and a computational point of view through practical examples. Networks with complex topological properties are a new discipline rapidly expanding due to its multidisciplinary nature:
it has found in fact applications in many fields, including finance, social sciences and biology.
The first part of the course is devoted to the characterization of the
topological structure of complex networks and to the study of the most used network models. The second part is focused on growth and dynamical processes in these systems and to the study of specific networks of this kind.
|
6
|
FIS/03
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410637 -
AM450 - FUNCTIONAL ANALYSIS
(objectives)
To acquire a good knowledge of functional analysis: Banach and Hilbert spaces, weak topologies, linear and continuous operators, compact operators, spectral theory.
|
9
|
MAT/05
|
48
|
24
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410690 -
MA410 - APPLIED AND INDUSTRIAL MATHEMATICS
(objectives)
Present a number of problems, of interest for application in various scientific and technological areas. Deal with the modeling aspects as well as those of numerical simulation, especially for problems formulated in terms of partial differential equations.
|
6
|
MAT/08
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410693 -
FM420 - Dynamic Systems
(objectives)
To acquire a solid knowledge on some advanced problems of interest in the theory of Dynamical Systems
|
6
|
MAT/07
|
48
|
12
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
20410441 -
CP420-Introduction to Stochastic Processes
(objectives)
Introduction to the theory of stochastic processes. Markov chains: ergodic theory, coupling, mixing times, with applications to random walks, card shuffling, and the Monte Carlo method. The Poisson process, continuous time Markov chains, convergence to equilibrium for some simple interacting particle systems.
|
6
|
MAT/06
|
-
|
-
|
-
|
-
|
Related or supplementary learning activities
|
ITA |
|
Università Roma Tre