Università Roma Tre

ACQUIRE GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF THE THEORY OF EQUATIONS IN ONE VARIABLE. UNDERSTAND AND BE ABLE TO APPLY THE “FUNDAMENTAL THEOREM OF CORRESPONDENCE OF GALOIS” IN ORDER STUDY COMPLEXITY OF A POLYNOMIAL.

TO OBTAIN A SOLID KNOWLEDGE OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS OF ONE COMPLEX VARIABLE AND OF THEIR MAIN PROPERTIES. TO DEVELOP PRACTICAL SKILLS IN THE USE OF COMPLEX FUNCTIONS, ESPECIALLY IN COMPLEX INTEGRATION AND IN COMPUTATION OF REAL DEFINITE INTEGRALS.

A REFINED STUDY OF TOPOLOGY VIA ALGEBRAIC AND ANALYTIC TOOLS.

To acquire deep understanding of the principal geometry arguments treated in high-school mathematics

To gain a solid knowledge of the basic aspects of probabilità theory: construction of probabilità measures on measurable spaces, 0-1 law, independence, conditional expectation, random variables, convergence of random variables, characteristic functions, central limit theorem, branching processes, discrete martingales.

THE PURPOSE OF THIS COURSE IS TO DEEPEN THE KNOWLEDGE OF SOME TOOLS AND FUNDAMENTAL PROPERTIES OF COMMUTATIVE RINGS AND THEIR MODULES, WITH PARTICULAR EMPHASIS TO THE CASE OF RINGS ARISING IN ALGEBRAIC NUMBER THEORY AND ALGEBRAIC GEOMETRY.

This course is an introduction to the methods and techniques of Algebraic Number Theory, with applications to the study of some classical problems.

INTRODUCTION TO THE ADVANCED THEORY OF MARKOV CHAINS, WITH SPECIAL EMPHASIS ON THE TOPIC OF CONVERGENCE TO EQUILIBRIUM AND ITS APPLICATIONS.

KNOWLEDGE ON OBJECT ORIENTED ANALYSIS AND DEVELOPMENT.
KNOWLWDGE ON UML MODELLING AND DESIGN AND ON WRITING COMPLEX ALGORITHMS IN JAVA.
KNOWLEDGE ON CONCURRENT PROGRAMMING IN JAVA.
BASIC KNOWLEDGE ON REMOTE AND DISTRIBUTED COMPUTING WITH JAVA AND ITS EXTENSIONS

Nowadays, information is often stored and transmitted electronically. Ensuring the security of the vast and complex infrastructure of computers, servers and networks is an immense challenge. In this scenario, this course aims at describing:
- The relationship between number theory and information security techniques;
- How to protect networks, secure electronic assets, prevent attacks, ensure the privacy of computer users, and build secure infrastructures.

Problem solving with examples from secondary school mathematical curricula, with the help of a computer and a direct use of numerical and symbolic calculus and dynamical geometry software (MATHEMATICA, introduction on CABRI and GEOGEBRA).
All examples, in an interactive and “laboratorial” lessons, point at experience limits and opportunities of using computers at school on selected arguments like numerical approximation or visualization in geometry as well as analysis.

The purpose of this course is to introduce mathematical statistics and statistical inference to students with good backgrounds in mathematics. Starting from the basic of probability this course, develop the theory of statistical inference using techniques, definition and concepts that are statistical and natural extensions and consequences of previous concepts. Purpose of this course it’s also to introduce students to the use of statistical software.

TO KNOW – AT A GRADUATE LEVEL – RESULTS ON LOGIC, INDEED MATHEMATICAL THEOREMS ON LOGIC, WHICH ARE AT THE BASIS OF THE INTERACTION BETWEEN LOGIC, COMPUTER SCIENCES AND OTHER SCIENCES (IN PARTICUALR, COMMUNICATION SCIENCES).

Application of the compactness theorem, Löwenheim-Skolem’s theorem. Basic recursion theory, decidability. Completeness and decidability of a first order theory, examples. Peano’s arithmetic and Gödel’s incompleteness theorems..

The axioms of Zermelo-Fraenkel. Odinal numbers. The axiom of foundation. The axiom of choice. The cardinal numbers and the continuum hypothesis.

GAIN KNOWLEDGE OF THE BASIC PRINCIPLES OF QUANTUM MECHANICS APPLIED TO SIMPLE PHYSICAL SYSTEMS

The course aims to improve the mathematical culture of the student and to give him further tools for preparing his specific final dissertation to obtain the degree.

The student is going to acquire the ability to read a scientific text in English, or , if he or she chooses, in any other European tongue.

The student will become familiar with the operation of computers: punched cards, magnetic tape data storage, ferrite core memory, etc.

LEARNING THE BASICS OF SPECTRAL THEORY FOR LINEAR OPERATORS LIMITED, WITH APPLICATIONS TO DIFFERENTIAL PROBLEMS.

Introduction to lebesgue theory and methods of linear functional analysis.

The aim of the course is to develop a good knowledge of fundamental methods in the solution of elementary problems in partial differential equations

THE COURSE IS INTENDED TO GIVE THE FUNDAMENTALS OF NUMERICAL APPROXIMATION TECHNIQUES, WITH A SPECIAL EMPHASIS ON THE SOLUTION OF LINEAR SYSTEMS AND NONLINEAR SCALAR EQUATIONS, POLYNOMIAL INTERPOLATION AND APPROXIMATE INTEGRATION FORMULAE. BESIDES BEING INTRODUCTORY, SUCH TECHNIQUES WILL BE USED IN THE SEQUEL AS BUILDING BLOCKS FOR MORE COMPLEX SCHEMES.

The course Theory of Computation and Interaction provides a in-deep view of theoretical aspects related to the concept of computation and the study of relations between different models of computation. The basic knowledge on information technology is here extended with new concepts and theoretical viewpoints. The course is divided into two units of 6 CFU. At choice, the student can decide to pass the first unit or both (12 CFU). More specifically, the course provides a formal presentation of the concepts of algorithm and computability. After the introduction of the classical concept of computability as formalized by Alan M. Turing, we address the basic concepts of algorithmic complexity and problem decidability, functional models and functional programming. In the second unit, we focus on interactive paradigms in the theory of computation which allow the description of additional complexity classes and their use in the semantics of programming languages.

TO ACQUIRE A GOOD KNOWLEDGE OF CONCEPTS AND
METHODS OF ELEMENTARY NUMBER THEORY, WITH PARTICULAR RESPECT OF STUDY OF DIOPHANTINE EQUATIONS AND POLYNOMIAL CONGRUENCES. TO GIVE PREREQUISITES FOR ADVANCED COURSES OF ALGEBRAIC AND ANALYTIC NUMBER THEORY.

THE COURSE PRESENTS A REVIEW OF NUMERICAL METHODS OF INCREASING IMPACT FOR APPLICATION. IN THIS LINE OF WORK, THE ELEMENTARY SCHEMES INTRODUCED IN THE FIRST COURSE ARE USED AS BUILDING BLOCKS FOR MORE COMPLEX METHODS, WITH THE FINAL GOAL OF INTRODUCING THE STUDENT (IN A SOMEWHAT SIMPLIFIED FRAMEWORK) TO THE GENERAL ASPECTS OF THE APPROXIMATE SOLUTION OF OPTIMIZATION PROBLEMS AND SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. ALL TECHNIQUES WILL BE TESTED ON SOME BENCHMARK PROBLEMS OF INTEREST FOR APPLICATION.

The student will learn the basic theorem of ODE theory.

ACQUIRE GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF PUBLIC KEY CRYPTOGRAPHY PROVIDING
AN OVERVIEW OF THOSE WHICH ARE CURRENTLY IN USE.

CONTINUING THE STUDY, BEGAN DURING FM1, OF DYNAMIC SYSTEMS OF PHYSICAL INTEREST WITH MOST STYLISH AND POWERFUL TECHNIQUES, SUCH AS THE LAGRANGIAN AND HAMITONIAN FORMALISM, THAT ARE IN THE VAST RANGE OF APPLICATIONS OF ANALYSIS AND MATHEMATICAL PHYSICS.

Introduction to the study of topological and geometrical structures defined using algebraic methods. Refinement of the algebraic knowledge usimg applications ti the study of algebraic varieties in affine and projective spaces.

WE STUDY THE GEOMETRY OF SURFACES IN EUCLIDEAN THREE-SPACE. THIS GIVES THE STUDENT CONCRETE EXAMPLES FOR THE NOTION OF CURVATURE IN GEOMETRY. WE USE METHODS OF MULTIVARIABLE CALCULUS, LINEAR ALGEBRA AND TOPOLOGY THUS GIVING THE STUDENTS A GOOD VIEW OF SOME ASPECTS OF MATHEMATICS. THE SOFTWARE “MATHEMATICA” IS ALSO USED TO VISUALIZE AND COMPUTE GEOMETRIC FEATURES OF SURFACES IN SPACE.

This course enables the student to:
1) Reach an understanding of the origin and evolution of mathematical thought in different historical and cultural contexts.
2) Consider the development of mathematics as a discipline and the relationship with philosophical thought, with the natural sciences and with technology and praxis.
3) Develop a cultural view of the role of mathematics in the contemporary world, including its transmission and matematical education.

A DEEP AND CRITICAL INTRODUCTION TO A SPECIFIC LOGICAL THEORY WHICH CONCERNS THE THEME OF THE INTERACTION AND IS CENTRAL IN THE CONTEMPORARY RESEARCH ON INFROMATION AND COMMUNICATION: LINEAR LOGIC AND ITS DEVELOPMENTS.

CALCULUS AND, IN PARTICULAR, DIFFERENTIAL EQUATIONS ARE IMPORTANT IN THE RESEARCH AND DEVELOPMENT OF APPLIED AND INDUSTRIAL MATEMATICS. THESE MATHEMATICAL TOOLS ARE NECESSARY TO UNDERSTAND A NUMBER OF PHYSICAL, CHEMICAL, BIOLOGICAL AND FINANCIAL PHENOMENA, AND TO IMPROVE THE QUALITY OF INDUSTRIAL PRODUCTS AND PROCESSES. MODELING AND SIMULATION COULD BE THE BEST TERMS TO DESCRIBE THE SPIRIT OF THIS COURSE. WE SHALL TRY TO CONVINCE THE PARTICIPANTS OF THE VALIDITY OF THE MATHEMATICAL APPROACH TO SOLVE CONCRETE PROBLEMS THROUGH THE PRESENTATION OF SOME CONCRETE EXAMPLES.

The course aims to improve the mathematical culture of the student and to give him further tools for preparing his specific final dissertation to obtain the degree.

The student is going to acquire the ability to read a scientific text in English, or , if he or she chooses, in any other European tongue.

The student will become familiar with the operation of computers: punched cards, magnetic tape data storage, ferrite core memory, etc.

The course provides a rigorous introduction to financial economic problems using mathematical methods, including the portfolio decision of an investor and the determination of the no-arbitrage price of stocks in both discrete and continuous time. The pricing of derivative securities in continuous time including various stock and interest rate options will also be included. Specific topics include derivative strategies, financial risk management techniques. The course aims to deepen the students' understanding of important concepts of mathematics of finance, the valuation of financial securities, capital investment evaluation and the estimation of required rates of return.

THE COURSE IS INTENDED TO REVIEW THE BASIC CONCEPTS IN THE NUMERICAL APPROXIMATION OF PARTIAL DIFFERENTIAL EQUATIONS (PDES), WITH RESPECT TO BOTH THEIR CONSTRUCTION (FINITE DIFFERENCES, FINITE ELEMENTS, SPECTRAL) AND THEIR CONVERGENCE ANALYSIS.

ACQUIRE GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF THE THEORY OF EQUATIONS IN ONE VARIABLE. UNDERSTAND AND BE ABLE TO APPLY THE “FUNDAMENTAL THEOREM OF CORRESPONDENCE OF GALOIS” IN ORDER STUDY COMPLEXITY OF A POLYNOMIAL.

TO OBTAIN A SOLID KNOWLEDGE OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS OF ONE COMPLEX VARIABLE AND OF THEIR MAIN PROPERTIES. TO DEVELOP PRACTICAL SKILLS IN THE USE OF COMPLEX FUNCTIONS, ESPECIALLY IN COMPLEX INTEGRATION AND IN COMPUTATION OF REAL DEFINITE INTEGRALS.

The course Theory of Computation and Interaction provides a in-deep view of theoretical aspects related to the concept of computation and the study of relations between different models of computation. The basic knowledge on information technology is here extended with new concepts and theoretical viewpoints. The course is divided into two units of 6 CFU. At choice, the student can decide to pass the first unit or both (12 CFU). More specifically, the course provides a formal presentation of the concepts of algorithm and computability. After the introduction of the classical concept of computability as formalized by Alan M. Turing, we address the basic concepts of algorithmic complexity and problem decidability, functional models and functional programming. In the second unit, we focus on interactive paradigms in the theory of computation which allow the description of additional complexity classes and their use in the semantics of programming languages.

To acquire deep understanding of the principal geometry arguments treated in high-school mathematics

To gain a solid knowledge of the basic aspects of probabilità theory: construction of probabilità measures on measurable spaces, 0-1 law, independence, conditional expectation, random variables, convergence of random variables, characteristic functions, central limit theorem, branching processes, discrete martingales.

THE PURPOSE OF THIS COURSE IS TO DEEPEN THE KNOWLEDGE OF SOME TOOLS AND FUNDAMENTAL PROPERTIES OF COMMUTATIVE RINGS AND THEIR MODULES, WITH PARTICULAR EMPHASIS TO THE CASE OF RINGS ARISING IN ALGEBRAIC NUMBER THEORY AND ALGEBRAIC GEOMETRY.

The student will learn the basic theorem of ODE theory.

INTRODUCTION TO THE ADVANCED THEORY OF MARKOV CHAINS, WITH SPECIAL EMPHASIS ON THE TOPIC OF CONVERGENCE TO EQUILIBRIUM AND ITS APPLICATIONS.

CONTINUING THE STUDY, BEGAN DURING FM1, OF DYNAMIC SYSTEMS OF PHYSICAL INTEREST WITH MOST STYLISH AND POWERFUL TECHNIQUES, SUCH AS THE LAGRANGIAN AND HAMITONIAN FORMALISM, THAT ARE IN THE VAST RANGE OF APPLICATIONS OF ANALYSIS AND MATHEMATICAL PHYSICS.

Introduction to the study of topological and geometrical structures defined using algebraic methods. Refinement of the algebraic knowledge usimg applications ti the study of algebraic varieties in affine and projective spaces.

WE STUDY THE GEOMETRY OF SURFACES IN EUCLIDEAN THREE-SPACE. THIS GIVES THE STUDENT CONCRETE EXAMPLES FOR THE NOTION OF CURVATURE IN GEOMETRY. WE USE METHODS OF MULTIVARIABLE CALCULUS, LINEAR ALGEBRA AND TOPOLOGY THUS GIVING THE STUDENTS A GOOD VIEW OF SOME ASPECTS OF MATHEMATICS. THE SOFTWARE “MATHEMATICA” IS ALSO USED TO VISUALIZE AND COMPUTE GEOMETRIC FEATURES OF SURFACES IN SPACE.

Nowadays, information is often stored and transmitted electronically. Ensuring the security of the vast and complex infrastructure of computers, servers and networks is an immense challenge. In this scenario, this course aims at describing:
- The relationship between number theory and information security techniques;
- How to protect networks, secure electronic assets, prevent attacks, ensure the privacy of computer users, and build secure infrastructures.

Problem solving with examples from secondary school mathematical curricula, with the help of a computer and a direct use of numerical and symbolic calculus and dynamical geometry software (MATHEMATICA, introduction on CABRI and GEOGEBRA).
All examples, in an interactive and “laboratorial” lessons, point at experience limits and opportunities of using computers at school on selected arguments like numerical approximation or visualization in geometry as well as analysis.

The purpose of this course is to introduce mathematical statistics and statistical inference to students with good backgrounds in mathematics. Starting from the basic of probability this course, develop the theory of statistical inference using techniques, definition and concepts that are statistical and natural extensions and consequences of previous concepts. Purpose of this course it’s also to introduce students to the use of statistical software.

TO KNOW – AT A GRADUATE LEVEL – RESULTS ON LOGIC, INDEED MATHEMATICAL THEOREMS ON LOGIC, WHICH ARE AT THE BASIS OF THE INTERACTION BETWEEN LOGIC, COMPUTER SCIENCES AND OTHER SCIENCES (IN PARTICUALR, COMMUNICATION SCIENCES).

Application of the compactness theorem, Löwenheim-Skolem’s theorem. Basic recursion theory, decidability. Completeness and decidability of a first order theory, examples. Peano’s arithmetic and Gödel’s incompleteness theorems..

GAIN KNOWLEDGE OF THE BASIC PRINCIPLES OF QUANTUM MECHANICS APPLIED TO SIMPLE PHYSICAL SYSTEMS

The course aims to improve the mathematical culture of the student and to give him further tools for preparing his specific final dissertation to obtain the degree.

The student is going to acquire the ability to read a scientific text in English, or , if he or she chooses, in any other European tongue.

The student will become familiar with the operation of computers: punched cards, magnetic tape data storage, ferrite core memory, etc.

LEARNING THE BASICS OF SPECTRAL THEORY FOR LINEAR OPERATORS LIMITED, WITH APPLICATIONS TO DIFFERENTIAL PROBLEMS.

Introduction to lebesgue theory and methods of linear functional analysis.

The aim of the course is to develop a good knowledge of fundamental methods in the solution of elementary problems in partial differential equations

A REFINED STUDY OF TOPOLOGY VIA ALGEBRAIC AND ANALYTIC TOOLS.

THE COURSE IS INTENDED TO GIVE THE FUNDAMENTALS OF NUMERICAL APPROXIMATION TECHNIQUES, WITH A SPECIAL EMPHASIS ON THE SOLUTION OF LINEAR SYSTEMS AND NONLINEAR SCALAR EQUATIONS, POLYNOMIAL INTERPOLATION AND APPROXIMATE INTEGRATION FORMULAE. BESIDES BEING INTRODUCTORY, SUCH TECHNIQUES WILL BE USED IN THE SEQUEL AS BUILDING BLOCKS FOR MORE COMPLEX SCHEMES.

TO ACQUIRE A GOOD KNOWLEDGE OF CONCEPTS AND
METHODS OF ELEMENTARY NUMBER THEORY, WITH PARTICULAR RESPECT OF STUDY OF DIOPHANTINE EQUATIONS AND POLYNOMIAL CONGRUENCES. TO GIVE PREREQUISITES FOR ADVANCED COURSES OF ALGEBRAIC AND ANALYTIC NUMBER THEORY.

THE COURSE PRESENTS A REVIEW OF NUMERICAL METHODS OF INCREASING IMPACT FOR APPLICATION. IN THIS LINE OF WORK, THE ELEMENTARY SCHEMES INTRODUCED IN THE FIRST COURSE ARE USED AS BUILDING BLOCKS FOR MORE COMPLEX METHODS, WITH THE FINAL GOAL OF INTRODUCING THE STUDENT (IN A SOMEWHAT SIMPLIFIED FRAMEWORK) TO THE GENERAL ASPECTS OF THE APPROXIMATE SOLUTION OF OPTIMIZATION PROBLEMS AND SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. ALL TECHNIQUES WILL BE TESTED ON SOME BENCHMARK PROBLEMS OF INTEREST FOR APPLICATION.

This course is an introduction to the methods and techniques of Algebraic Number Theory, with applications to the study of some classical problems.

ACQUIRE GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF PUBLIC KEY CRYPTOGRAPHY PROVIDING
AN OVERVIEW OF THOSE WHICH ARE CURRENTLY IN USE.

KNOWLEDGE ON OBJECT ORIENTED ANALYSIS AND DEVELOPMENT.
KNOWLWDGE ON UML MODELLING AND DESIGN AND ON WRITING COMPLEX ALGORITHMS IN JAVA.
KNOWLEDGE ON CONCURRENT PROGRAMMING IN JAVA.
BASIC KNOWLEDGE ON REMOTE AND DISTRIBUTED COMPUTING WITH JAVA AND ITS EXTENSIONS

This course enables the student to:
1) Reach an understanding of the origin and evolution of mathematical thought in different historical and cultural contexts.
2) Consider the development of mathematics as a discipline and the relationship with philosophical thought, with the natural sciences and with technology and praxis.
3) Develop a cultural view of the role of mathematics in the contemporary world, including its transmission and matematical education.

A DEEP AND CRITICAL INTRODUCTION TO A SPECIFIC LOGICAL THEORY WHICH CONCERNS THE THEME OF THE INTERACTION AND IS CENTRAL IN THE CONTEMPORARY RESEARCH ON INFROMATION AND COMMUNICATION: LINEAR LOGIC AND ITS DEVELOPMENTS.

The axioms of Zermelo-Fraenkel. Odinal numbers. The axiom of foundation. The axiom of choice. The cardinal numbers and the continuum hypothesis.

CALCULUS AND, IN PARTICULAR, DIFFERENTIAL EQUATIONS ARE IMPORTANT IN THE RESEARCH AND DEVELOPMENT OF APPLIED AND INDUSTRIAL MATEMATICS. THESE MATHEMATICAL TOOLS ARE NECESSARY TO UNDERSTAND A NUMBER OF PHYSICAL, CHEMICAL, BIOLOGICAL AND FINANCIAL PHENOMENA, AND TO IMPROVE THE QUALITY OF INDUSTRIAL PRODUCTS AND PROCESSES. MODELING AND SIMULATION COULD BE THE BEST TERMS TO DESCRIBE THE SPIRIT OF THIS COURSE. WE SHALL TRY TO CONVINCE THE PARTICIPANTS OF THE VALIDITY OF THE MATHEMATICAL APPROACH TO SOLVE CONCRETE PROBLEMS THROUGH THE PRESENTATION OF SOME CONCRETE EXAMPLES.

The course aims to improve the mathematical culture of the student and to give him further tools for preparing his specific final dissertation to obtain the degree.

The student is going to acquire the ability to read a scientific text in English, or , if he or she chooses, in any other European tongue.

The student will become familiar with the operation of computers: punched cards, magnetic tape data storage, ferrite core memory, etc.

The course provides a rigorous introduction to financial economic problems using mathematical methods, including the portfolio decision of an investor and the determination of the no-arbitrage price of stocks in both discrete and continuous time. The pricing of derivative securities in continuous time including various stock and interest rate options will also be included. Specific topics include derivative strategies, financial risk management techniques. The course aims to deepen the students' understanding of important concepts of mathematics of finance, the valuation of financial securities, capital investment evaluation and the estimation of required rates of return.

THE COURSE IS INTENDED TO REVIEW THE BASIC CONCEPTS IN THE NUMERICAL APPROXIMATION OF PARTIAL DIFFERENTIAL EQUATIONS (PDES), WITH RESPECT TO BOTH THEIR CONSTRUCTION (FINITE DIFFERENCES, FINITE ELEMENTS, SPECTRAL) AND THEIR CONVERGENCE ANALYSIS.