Applied Mathematics

Educational objectives

General targets:

To acquire knowledge in numerical linear algebra and numerical modeling for differential problems

Specific targets:

Knowledge and understanding: At the end of the Course students will have theoretical knowledge related to methods of numerical analysis for the solution of linear systems and eigenvalue problems and for the integration of ordinary differential equations and linear partial differential equations. Also, they will have acquired techniques related to implementation of algorithms for the effective solution of the problems.

Applying knowledge and understanding: Students who have passed the exam will be able to use methodologies for the numerical solution of a linear system or of an eigenvalue problem and for the discretization of ordinary differential equations or linear partial derivatives. Also, they will be able to predict performance of such algorithms depending on the characteristics of the problem to deal with.

Making judgements: Students who have passed the exam will be able to select, among the algorithms that they will have studied during the Course, those suited to the solution of the problem to be treated, being also able to make the modifications that may be necessary to improve their performance.

Communication skills: Students will have gained the ability to communicate concepts, ideas and methodologies of numerical linear algebra and numerical modeling for differential problems.

Learning skills: The acquired knowledge will allow students who have passed the exam to face the study, at the individual level or in a Master's degree course, of more specialized aspects of numerical linear algebra and numerical modeling for differential problems, being able to understand the specific terminology and identify the most relevant topics.

Educational objectives

General objectives: to acquire basic knowledge in elementary theory of numbers and finite fields (useful when studying public key cryptography or code theory in other courses or contexts).
 
Specific objectives:
 
Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the elementary theory of numbers, the resolution of equations to the comparisons (with particular regard to polynomial equations), akke arithmetic functions, the theory of quadratic residues, to the problem of enlargements, extensions and the detailed structure of finite fields.
 
Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of techniques related to equations to congruences, to the most important arithmetic functions; will be able to describe in a concrete way a finite field and his group of Galois.
 
Critical and judgmental skills: the student will have the basics to use the tools that underlie public-key cryptography and the theory of error-correcting codes. The aim of the course, which is purely theoretical, is to allow students interested in the cryptographic applications to easily manipulate the mathematical objects of the course.
 
Communication skills: ability to expose the contents in the oral part of the verification and in any theoretical questions present in the written test.
 
Learning skills: the acquired knowledge will allow a study, individual or given in an LM course, related to standard aspects of number theory and finite fields.

Educational objectives

General objectives of the course: The course has two objectives, one for each module;
1) To provide the algebraic-topological basis to understand the techniques of topological data analysis.
2) Deepen some aspects of classical projective geometry and theory of algebraic curves, usually neglected in the three-year degree courses.

Educational objectives

Knowledge and understanding: at the end of the course the student will have acquired the notions and the results
basics related to singular homology, to the study of differentiable varieties
and a fair knowledge of Riemann's theory of surfaces.

Apply knowledge and understanding: at the end of the course the student will be able to solve
even complex problems that require the use of techniques related to de Rham's cohomology,
the Hurwitz theorem and the Riemann Roch theorem for compact Riemann surfaces;
will be able to determine the kind of a Riemann Surface and the size of the linear systems
variety cohomology groups.

Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between
topics that vary between algebraic topology, differential geometry, complex geometry
and also algebraic geometry.

Communication skills: ability to expose the contents in the oral part of the verification and in the
any theoretical questions present in the written test.

Learning skills: the knowledge acquired will allow you to devote yourself to more specialized aspects of geometry.

Educational objectives

Educational Goals

General objectives:

Acquire basic knowledge in modeling based on ordinary and partial differential equations, in the contexts presented in the program. In particular, he will be able to treat differential equations for networks of chemical reactions, the spread of epidemics, the kinetics of enzymes, the propagation of nerve impulses; in addition, he will be able to deal with models in which there is also dependence on space with diffusive terms.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results relating to some classes of ordinary differential equations and partial derivative equations useful for the description of models, mainly in the biochemical and epidemiological fields.

Apply knowledge and understanding: at the end of the course the student will be able to present basic models in the biomathematic field, discussing their properties and characteristics. You will also be able to use the electronic calculator to perform basic numerical simulations of nonlinear differential equations using pre-existing libraries.

Critical and judgmental skills: the student will have the bases to analyze the analogies and relationships between the topics covered and topics acquired in previous courses in the same field, critically recognizing their salient features.

Communication skills: the student will have developed the ability to expose the contents in the oral part of the verification.

Learning skills: the knowledge acquired will allow an individual and collective study of the subsequent LM courses that require modeling skills.

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

General objectives: Introduce the student to the fundamental results of mathematical statistics and to the most significant applications, also through the discussion of concrete cases and statistical software.
 
Specific objectives:
 
Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results concerning the problems of punctual estimation, by interval and the problems of hypothesis testing, as well as the main methods with which they are faced: method of moments, of the maximum likelihood and generalizations.
 
Apply knowledge and understanding: at the end of the course the student will be able to assess the degree of accuracy with which, in simple statistical problems, parameters can be estimated or validated on these, implementing these responses in an appropriate software.
 
Critical and judgmental skills: the student will be able to appreciate the probabilistic tools useful for dealing with statistical problems and the various approaches to resolving them.
 
Communication skills: ability to expose the contents in the oral part of the assessment and in any theoretical questions present in the written test.
 
Learning skills: the acquired knowledge will allow a subsequent study of more recent and advanced aspects of mathematical statistics.

Educational objectives

General objectives: to acquire basic knowledge in stochastic process theory and in stochastic modeling
of real phenomena.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results concerning stochastic processes in discrete and continuous time, on discrete structures such as graphs or on continuous spaces.

Apply knowledge and understanding: at the end of the course the student will be able to model the temporal evolution of various real phenomena through stochastic processes, to analyze the stationarity and / or temporal reversibility of stochastic processes, to calculate probabilities of absorption and expected absorption times, to simulate stochastic processes and to estimate the rate of convergence at equilibrium.

Critical and judgmental skills: the student will have the basis to study stochastic dynamic systems and acquire the ability to evaluate the goodness of a model compared to others in the modeling of real phenomena.

Communication skills: having to take an oral theory test, students will develop the communication skills necessary to expose the mathematical theory and the various models considered in the course.

Learning skills: the acquired knowledge will allow a more in-depth study of stochastic processes both on discrete and continuous spaces, helping the student to study other courses such as stochastic calculus.

Educational objectives

General targets: To acquire advanced knowledge in the theory of dynamical systems.

Specific targets:

Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on applications in mechanics and applied sciences in general. They will learn elements of stability theory and hyperbolic theory (such as homoclinic intersections and existence of chaotic motions). They will also learn elements of the theory of topological dynamical systems and of ergodic theory.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study stability problems of equilibria and cycles, both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.

Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.

Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.

Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.

Educational objectives

Knowledge and understanding:The course gives to successful students some advanced tools for the study of various linear and nonlinear PDE's. They will reach a good familiarity with the most recent notions of solutions and their qualitative properties.Skills and attributes:Successful students will able to deal with the advanced study of the solutions to various types of linear and nonlinear PDE's.

Educational objectives

General skills
This course is designed to explore the fundamentals of time series analysis.

Specific skills

Knowledge and understanding:
Knowing and understanding of basic results of mathematical models of time series: stationary and non stationary processes, multivariate linear models, ARIMA models, spectral analysis, trend, test of serial independance.

Applying knowledge and understanding:
Be able to analyze simple data series, to estimate parameters, to extract trend and noise, to perform residual diagnostics.

Making judgements:
Be able to understand relationship to basic linear algebra, analysis, probability and statistics.

Communication skills:
Be able to communicate what has been learned during the laboratory and oral exam.

Learning skills:
Be able to learning the specific terminology and advanced methods on time series.

Educational objectives

The course is devoted to the study of multiscale approaches (micro-meso-macro) for multi-agent systems. Typical examples are: vehicular traffic, crowd dynamics, opinion dynamics, flocking/swarming, financial markets and so on.
The course includes lab sessions for the computational part related to the numerica simulation of the models.

1. Knowledge and understanding

Students who have passed the exam will know how to model and study qualitative properties of physical phenomena through several scales of representation: from the microscopic, to the kinetic and the macroscopic one.

2. Applied knowledge and understanding

Students who have passed the exam will be able to use a efficient numerical techniques, deterministic and not, for the simulation of models, and they will be able to code the algorithms in C++ or MATLAB.

3. Making judgments

Students will be able to evaluate the right representation scale of the given phenomenon, the results produced by their programs and to produce tests and simulations.

4. Communication skills

Students will be able to present and explain the modeling choices, the properties of the models, either at the blackboard and/or using a computer.

5. Learning skills

The acquired knowledge will construct the basis to study more research topics related to the modeling of multi-agent systems.

Educational objectives

General targets:
to acquire basic and advanced knowledge and hands-on experience on
some classic topics in finite-dimensional optimization.

Specific targets

Knowledge and understanding:
Understanding of the theoretical foundations of optimization theory and of the main
algorithm classes for the solution of optimization problems.

Applying knowledge and understanding:
the student will be able to identify relevant characteristics of optimization problems
and to select the most appropriate solution method for a given problem, also taking into account practical constraints due to the applicative environment (for example, the required accuracy or time limits). In addition the student will be able to correctly
analyze the results provided by commercial or ad-hoc resolution software.

Making judgements:
ability to enucleate the most significant aspects of an optimization problem and of its solution methods.

Communication skills:
ability to enucleate the significant points of the theory,
to know how to illustrate the most interesting parts with appropriate examples,
to discuss mathematically the most subtle points.

Learning skills:
the acquired knowledge will allow the student to undertake
more advanced studies in optimization and to be able to work
in industrial and research environments where optimization is used.

Educational objectives

1) Knowledge and understanding
At the end of the course, the students will know and understand:
a) the idea of control system and of differential inclusion, and their basic properties;
b) thr idea of optimal control and necessary and/or sufficient conditions for its existence;
c) the relationship between optimal solutions of a control problem and the Hamilton-Jacobi-Bellman equation;
d) the idea of viscosity solution for the Hamilton-Jacobi equation.

2) Applying knowledge and understanding
At the end of the course, the students will be able to:
a) write the mathematical formulation of an optimal control problem;
b) determine, using the Pontryagin Maximum Principle, the optimal solutions of an optimal control problem;
c) analyze, from a theoretical point of view, the solutions of an optimal control problem through the study of the associated Hamilton-Jacobi-Bellman equation.

3) Making judgements
During the lessons, several problems will be proposed to the students.
Thanks to the autonomous resolution of the problems, and the subsequent discussion in the classroom, the students will acquire both the ability to evaluate their knowledge and the ability to tackle a wide range of optimal control problems.

4) Communication skills
The written form of the exercises, assigned either during lessons or during the written test, and the oral exam will allow the students to evaluate their skill in correctly communicating the knowledges acquired during the course.

5) Learning skills
At the end of the course the students will be able to analyze optimal control problems; such skill is acquired by means of several model problems assigned during the course.

Educational objectives

The course will present the fundamental results related to the analysis and approximation of scalar conservation laws and Hamilton-Jacobi equations. Moreover the course will illustrate a number of models leading to these equations: gas dynamics, traffic models on networks, optimal control problems, image processing, front propagation.

The course includes some Lab sessions to develop programming codes in C++ or MATLAB.

Knowledge and understanding:
Students who have passed the exam will know the main numerical techniques on the topics presented in the course.

Applied knowledge and understanding:
Students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to implement the algorithms in C++ or MATLAB.

Critical and judgmental skills:
Students will be able to evaluate the results produced by their programs and to produce tests and simulations.

Communication skills:
Students will be able to expose and motivate the proposed solution of some problems chosen in class either on the blackboard and/or using a computer.

Learning skills:
The acquired knowledge will allow to build the bases for a study related to more specialized aspects of the analysis and approximation of non linear partial differential equations. The student will become familiar with different concepts and techniques related to the topics presented in the course.

Educational objectives

General targets:
acquire basic knowledge of the physical and mathematical aspects of Fluid Mechanics and Kinetic Theory.

Knowledge and understanding:
knowledge of physical principles and modeling assumptions that lead to the equations of fluids and particle systems;
knowledge of fluid and gas equations and their mathematical properties:
weak formulations, existence and uniqueness of solutions,
models for the evolution of singular data.

Applying knowledge and understanding:
the student will be able to
modeling fluid and particle motions, also through the formulation of appropriate
action functionals, discuss the evolution of singularity, use the mathematical tools for
the treatment of fluids and gases in other contexts.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises.

Making judgements:
ability to identify the most significant aspects of the theory,
to know how to evaluate the limits and the advantages of simplifications
operated (incompressibility, absence or presence of viscosity...),
and the limits of mathematical results.

Communication skills:
ability to expose the development of the physical-mathematical
theory for fluids and particle systems, highlighting
the relationship between physical and mathematical aspects;
ability to illustrate the demonstrations,
summarizing the main ideas, and discussing the mathematical details.

Communication skills:
the acquired knowledge will allow a study, individual or given in an LM course, related to numerical aspects
or modeling of fluid mechanics and kinetic theory.

Educational objectives

General skills

The course aims to transmit to students a deep knowledge of the mathematical structure of Quantum Mechanics, of the historical and conceptual path leading to its formulation, and of its relations with other mathematical subjects (as e.g. functional analysis, operator theory, theory of Lie groups and their unitary representations).

Specific skills

A) Knowledge and understanding
After the conclusion of the course, successful students will know and understand the fundamental concepts of Fourier theory, the mathematical analogy between classical mechanics and geometric optics, the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics, and the mathematical structure of Quantum Theory, with a particular emphasis on dynamical aspects (time evolution) and on the analysis of the symmetries of a quantum system (representation of the symmetry group).

B) Applying knowledge and understanding
The general knowledge will be complemented by the application of general concepts to some specific models, and by the ability to analyze symmetries and dynamics of simple quantum systems. Specific simple systems will be analyzed in detail, including the case of a quantum particle in a linear potential, in a harmonic potential, in a uniform magnetic field, and in a Kepler potential (hydrogenoid atom). Successful students will be potentially able to apply the general concepts also to other more complex systems, including non-hydrogenoid atoms, molecules and crystalline solids.

C) Making judgements
The analysis of the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics will make successful students able to autonomously judge the epistemological foundations of a physical theory, and hence to understand its natural range of application and validity. This critical judgement will lead students to privilege an epistemological apophantic approach, with respect to an apodictic one.
Moreover, successful students will be able to autonomously judge the validity of a mathematical statement, through a critical analysis of the hypotheses and of the deductive steps leading to the proof of the statement itself, and to autonomously formulate counterexamples to mathematical statements whenever one of the hypotheses is denied.

D) Communication skills
Successful students will acquire the ability to communicate what has been learned through written themes and oral exams, and to formulate a logically structured speech, with a clear distinction between hypotheses, deduction and thesis.

E) Learning skills
Successful students will acquire the ability to identify the most relevant topics in a subject and to make the logical connections between the topics covered.

Educational objectives

General objective : The main purpose of the course is to give the student a good knowledge of the basic topics in Nonlinear Analysis which are important in the study of Differential Equations.

Specific objectives :
Knowledge and understanding: at the end of the course the student will have learned the basic theory to study differential problems with a variational structure, in particular those involving semilinear elliptic equations.
Applications : at the end of the course the student will be able to solve simple problems which require the use of variational methods to study critical points of nonlinear functionals.
Critical abilities: the student will have the basic knowledge of the variational theory of Differential Equations. He/she will be able to choose the appropriate methods to study nonlinear differential problems.
Communication skills: the student will have the ability to expose the topics studied in the oral exam.
Learning skills: the student will be capable to face the study of nonlinear variational problems which arise in the field of Differential Equations so that he/she can continue the study of more advanced topics.

Educational objectives

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

Educational objectives

Knowledge and understanding:Successful students will learn various characterizations of Brownianan motion, the fundamental properties of diffusion processes and the main results of stochastic calculus, including the Ito formula.Skills and attributes:Successful students will be able to apply stochastic calculus in various applications, from mathematical finance to physics and biology.

Educational objectives

General objectives

Acquiring basic knowledge on the mathematical methods used in artificial intelligence modeling, with particular attention to "machine learning".

Specific objectives

Knowledge and understanding: at the end of the course the student will have knowledge of the basic notions and results (mainly in the areas of stochastic processes and statistical mechanics) used in the study of the main models of neural networks (e.g., Hopfield networks, Boltzmann machines, feed-forward networks).

Apply knowledge and understanding: the student will be able to identify the optimal architecture for a certain task and to solve the resulting model by determining a phase diagram; the student will have the basis to independently develop algorithms for learning and retrieval.

Critical and judgmental skills: the student will be able to determine the parameters that control the qualitative behaviour of a neural network and to estimate the values of these parameters that allow a good performance of the network; she/he will also be able to investigate the analogies and relationships between the topics covered during the course and during courses dedicated to statistics and data analysis.

Communication skills: ability to expose the contents in the oral and written part of the verification, possibly by means of presentations.

Learning skills: the knowledge acquired will allow a study, individual or taught in a LM course, related to more specialised aspects of statistical mechanics, development of algorithms, usage of big data.

Educational objectives

The course aims to introduce students to the theory of viscosity solutions and to the metric and variational aspects of first-order Hamilton-Jacobi equations (weak KAM Theory) and to present some applications to asymptotic problems.

1. Knowledge and understanding.

At the end of the lectures the student will be familiar with the basic notions and results of the theory of viscosity solution and with the metric and variational aspects of first-order HJ equations (weak KAM Theory).

2. Applied knowledge and understanding.

Students who have passed the exam will be able to derive explicit expressions for solutions of first-order HJ equations in some simple examples and to derive qualitative information in more general cases.

3. Making judgments.

The students will acquire a satisfactory knowledge of the main tools and results of weak KAM Theory, which will provide them of a valuable insight on the geometric and dynamical phenomena taking place in the study of first-order HJ equations.

4. Communication skills

Ability to present the content during the oral exam.

5. Learning skills

Students will acquire the necessary tools to face the study of first-order Hamilton-Jacobi equations and to possibly approach research topics.

Educational objectives

General objectives: to acquire knowledge in High dimensional Probability and Statistics with applications to Data Science

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions of High Dimensional Probability and Statistics and will be familiar with algorithms used to solve some relevant problems in Data Science.

Apply knowledge and understanding: at the end of the course the student will be able to solve some problems concerning high dimensional random geometric structures, data dimension reduction, statistical learning and high dimensional regression

Critical and judgmental skills: the student will realize the ideas behind several algorithms and software used in Data Science,
understand optimal conditions and/or possible limits for applications

Communication skills: the student must show the ability to present the contents of the course in the oral part of the assessment and in the solution of problems in the written test.

Learning skills: the acquired knowledge will allow a multidisciplinary understanding of several problems motivated by data science and will facilitate the study into some very active research fields.

Educational objectives

Educational Goals

General objectives:

Acquire basic knowledge in modeling based on ordinary and partial differential equations, in the contexts presented in the program. In particular, he will be able to treat differential equations for networks of chemical reactions, the spread of epidemics, the kinetics of enzymes, the propagation of nerve impulses; in addition, he will be able to deal with models in which there is also dependence on space with diffusive terms.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results relating to some classes of ordinary differential equations and partial derivative equations useful for the description of models, mainly in the biochemical and epidemiological fields.

Apply knowledge and understanding: at the end of the course the student will be able to present basic models in the biomathematic field, discussing their properties and characteristics. You will also be able to use the electronic calculator to perform basic numerical simulations of nonlinear differential equations using pre-existing libraries.

Critical and judgmental skills: the student will have the bases to analyze the analogies and relationships between the topics covered and topics acquired in previous courses in the same field, critically recognizing their salient features.

Communication skills: the student will have developed the ability to expose the contents in the oral part of the verification.

Learning skills: the knowledge acquired will allow an individual and collective study of the subsequent LM courses that require modeling skills.

Educational objectives

Educational Goals

General objectives: To provide students with the basics related to the study of functional spaces that intervene in various fields. In particular, linear operators will be studied between Banach or Hilbert spaces and their spectrum will be analyzed. Finally, some non-linear Functional Analysis techniques will be presented, suitable for the study of differential problems.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the Functional Analysis and to its different applications to differential problems.

Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of functional analysis techniques.

Critical and judgmental skills: the student will have the bases to analyze the similarities and relationships between the topics covered and topics already seen in previous courses; will also acquire the tools that have historically led to the solution of classical problems. It will be able (at least in model cases) to recognize functional spaces suitable for solving analysis problems, for example differential problems with boundary conditions.

Communication skills: ability to expose the contents in the oral part of the assessment and in any theoretical questions present in the written test.

Learning skills: the acquired knowledge will allow the application of the techniques to advanced problems of Functional Analysis and to differential problems.

Educational objectives

General objectives:

Many models in mathematical physics and natural sciences in general have variational principles (the principle of minimal energy, minimal action, ...) which describe their equilibrium configurations and dynamic evolutions.
The aim of the course is to make students aware of the variety of problems that can be addressed with variational techniques and to provide them with the basic tools and mathematical language for analyzing the models arising in natural sciences.

Specific objectives:

Knowledge and understanding:

at the end of the course the student will have acquired the basic notions and results on the direct method of calculus variations, conditions for semicontinuity, asymptotic analysis via Gamma convergence, and she/he will be able to apply this methods in various contexts about which they will be provided the functional bases at least in dimension 1 (integral functionals and Sobolev spaces, geometric functionals and elements of geometric measure theory).

Apply knowledge and understanding:

at the end of the course the student will be able to begin the study of advanced calculus of variations. She/he will also be able to formulate a simple variational model (for example linked to a specific application) and analyze its asymptotic behavior or identify the characteristics that make it a robust model.

Critical and judgmental skills:

The student will have the basics to connect and use tools covered in various moments of his studies ranging from analysis, mathematical physics and to probability. She/he will therefore be able to appreciate the interest of a mathematical question in relation also to its use to answer a question coming from an applied problem.

Communication skills:

ability to rigorously expose the theoretical contents of the course and also ability to formulate the problem under consideration by understanding the role of deriving the right model and its analysis. Ability to explain moreover the results in the language related to the application under consideration, potentially understandable by non expert in calculating variations.

Learning ability:

the acquired knowledge will allow to face a possible master's thesis work in the field of applied mathematics in natural sciences both with a more theoretical approach and in connection with the analysis of a specific model of interest for applications.

Educational objectives

Educational Goals

General objectives:

Acquire basic knowledge in modeling based on ordinary and partial differential equations, in the contexts presented in the program. In particular, he will be able to treat differential equations for networks of chemical reactions, the spread of epidemics, the kinetics of enzymes, the propagation of nerve impulses; in addition, he will be able to deal with models in which there is also dependence on space with diffusive terms.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results relating to some classes of ordinary differential equations and partial derivative equations useful for the description of models, mainly in the biochemical and epidemiological fields.

Apply knowledge and understanding: at the end of the course the student will be able to present basic models in the biomathematic field, discussing their properties and characteristics. You will also be able to use the electronic calculator to perform basic numerical simulations of nonlinear differential equations using pre-existing libraries.

Critical and judgmental skills: the student will have the bases to analyze the analogies and relationships between the topics covered and topics acquired in previous courses in the same field, critically recognizing their salient features.

Communication skills: the student will have developed the ability to expose the contents in the oral part of the verification.

Learning skills: the knowledge acquired will allow an individual and collective study of the subsequent LM courses that require modeling skills.

Educational objectives

Educational Goals

General objectives: To provide students with the basics related to the study of functional spaces that intervene in various fields. In particular, linear operators will be studied between Banach or Hilbert spaces and their spectrum will be analyzed. Finally, some non-linear Functional Analysis techniques will be presented, suitable for the study of differential problems.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the Functional Analysis and to its different applications to differential problems.

Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of functional analysis techniques.

Critical and judgmental skills: the student will have the bases to analyze the similarities and relationships between the topics covered and topics already seen in previous courses; will also acquire the tools that have historically led to the solution of classical problems. It will be able (at least in model cases) to recognize functional spaces suitable for solving analysis problems, for example differential problems with boundary conditions.

Communication skills: ability to expose the contents in the oral part of the assessment and in any theoretical questions present in the written test.

Learning skills: the acquired knowledge will allow the application of the techniques to advanced problems of Functional Analysis and to differential problems.

Educational objectives

General objectives:

Many models in mathematical physics and natural sciences in general have variational principles (the principle of minimal energy, minimal action, ...) which describe their equilibrium configurations and dynamic evolutions.
The aim of the course is to make students aware of the variety of problems that can be addressed with variational techniques and to provide them with the basic tools and mathematical language for analyzing the models arising in natural sciences.

Specific objectives:

Knowledge and understanding:

at the end of the course the student will have acquired the basic notions and results on the direct method of calculus variations, conditions for semicontinuity, asymptotic analysis via Gamma convergence, and she/he will be able to apply this methods in various contexts about which they will be provided the functional bases at least in dimension 1 (integral functionals and Sobolev spaces, geometric functionals and elements of geometric measure theory).

Apply knowledge and understanding:

at the end of the course the student will be able to begin the study of advanced calculus of variations. She/he will also be able to formulate a simple variational model (for example linked to a specific application) and analyze its asymptotic behavior or identify the characteristics that make it a robust model.

Critical and judgmental skills:

The student will have the basics to connect and use tools covered in various moments of his studies ranging from analysis, mathematical physics and to probability. She/he will therefore be able to appreciate the interest of a mathematical question in relation also to its use to answer a question coming from an applied problem.

Communication skills:

ability to rigorously expose the theoretical contents of the course and also ability to formulate the problem under consideration by understanding the role of deriving the right model and its analysis. Ability to explain moreover the results in the language related to the application under consideration, potentially understandable by non expert in calculating variations.

Learning ability:

the acquired knowledge will allow to face a possible master's thesis work in the field of applied mathematics in natural sciences both with a more theoretical approach and in connection with the analysis of a specific model of interest for applications.

Educational objectives

General objectives:
acquire basic knowledge in Riemannian geometry.

Specific objectives:

Knowledge and understanding:
at the end of the course the student will have acquired the basic notions and results relating to the Riemannian varieties, connections and the different notions of curvature, the geodesics and fields of Jacobi, completeness and spaces with constant curvature.

Apply knowledge and understanding:
at the end of the course the student will be able to begin the study of advanced topics of Riemannian geometry, and to solve complex problems in this area.

Critical and judgmental skills:
the student will have the basis to analyze and appreciate the analogies and connections between the topics covered and the most varied themes coming from differential, algebraic topology, from algebraic and complex geometry.

Communication skills:
ability to rigorously expose the contents in the most theoretical questions present in the written test, and in the eventual oral part of the verification.

Learning ability:
the acquired knowledge will allow to face a possible master's thesis work on advanced topics of differential / Riemannian geometry, but also of complex analytical / differential geometry.

Educational objectives

Knowledge and understanding:The course gives to successful students some advanced tools for the study of various linear and nonlinear PDE's. They will reach a good familiarity with the most recent notions of solutions and their qualitative properties.Skills and attributes:Successful students will able to deal with the advanced study of the solutions to various types of linear and nonlinear PDE's.

Educational objectives

1) Knowledge and understanding
At the end of the course, the students will know and understand:
a) the idea of control system and of differential inclusion, and their basic properties;
b) thr idea of optimal control and necessary and/or sufficient conditions for its existence;
c) the relationship between optimal solutions of a control problem and the Hamilton-Jacobi-Bellman equation;
d) the idea of viscosity solution for the Hamilton-Jacobi equation.

2) Applying knowledge and understanding
At the end of the course, the students will be able to:
a) write the mathematical formulation of an optimal control problem;
b) determine, using the Pontryagin Maximum Principle, the optimal solutions of an optimal control problem;
c) analyze, from a theoretical point of view, the solutions of an optimal control problem through the study of the associated Hamilton-Jacobi-Bellman equation.

3) Making judgements
During the lessons, several problems will be proposed to the students.
Thanks to the autonomous resolution of the problems, and the subsequent discussion in the classroom, the students will acquire both the ability to evaluate their knowledge and the ability to tackle a wide range of optimal control problems.

4) Communication skills
The written form of the exercises, assigned either during lessons or during the written test, and the oral exam will allow the students to evaluate their skill in correctly communicating the knowledges acquired during the course.

5) Learning skills
At the end of the course the students will be able to analyze optimal control problems; such skill is acquired by means of several model problems assigned during the course.

Educational objectives

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

Educational objectives

General aim: to acquire a basic knowledge of Algebraic Geometry over an arbitrary ring.

Knowledge and understanding: at the end of the course the student will be acquainted with basic notions and results in the theory of schemes, cohomology of coherent sheaves, and the theory of projective curves and surfaces.

Applying knowledge and understanding: at the end of the course the student will be able to read and comprehend some papers in Algebraic Geometry.

Analytical and judgment abilities: the student will appreciate the analogies between classical Algebraic Geometry and Number Theory.

Communication skills: the student will be able to communicate the contents of the lectures, in particular illustrating them via concrete examples.

Learning skills: the acquired notions will allow the student to study (either by themselves or in a PhD course) more advanced topics in Algebraic Geometry.

Educational objectives

General objectives: to acquire basic knowledge in stochastic process theory and in stochastic modeling
of real phenomena.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results concerning stochastic processes in discrete and continuous time, on discrete structures such as graphs or on continuous spaces.

Apply knowledge and understanding: at the end of the course the student will be able to model the temporal evolution of various real phenomena through stochastic processes, to analyze the stationarity and / or temporal reversibility of stochastic processes, to calculate probabilities of absorption and expected absorption times, to simulate stochastic processes and to estimate the rate of convergence at equilibrium.

Critical and judgmental skills: the student will have the basis to study stochastic dynamic systems and acquire the ability to evaluate the goodness of a model compared to others in the modeling of real phenomena.

Communication skills: having to take an oral theory test, students will develop the communication skills necessary to expose the mathematical theory and the various models considered in the course.

Learning skills: the acquired knowledge will allow a more in-depth study of stochastic processes both on discrete and continuous spaces, helping the student to study other courses such as stochastic calculus.

Educational objectives

General targets: To acquire advanced knowledge in the theory of dynamical systems.

Specific targets:

Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on applications in mechanics and applied sciences in general. They will learn elements of stability theory and hyperbolic theory (such as homoclinic intersections and existence of chaotic motions). They will also learn elements of the theory of topological dynamical systems and of ergodic theory.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study stability problems of equilibria and cycles, both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.

Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.

Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.

Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

The course will present the fundamental results related to the analysis and approximation of scalar conservation laws and Hamilton-Jacobi equations. Moreover the course will illustrate a number of models leading to these equations: gas dynamics, traffic models on networks, optimal control problems, image processing, front propagation.

The course includes some Lab sessions to develop programming codes in C++ or MATLAB.

Knowledge and understanding:
Students who have passed the exam will know the main numerical techniques on the topics presented in the course.

Applied knowledge and understanding:
Students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to implement the algorithms in C++ or MATLAB.

Critical and judgmental skills:
Students will be able to evaluate the results produced by their programs and to produce tests and simulations.

Communication skills:
Students will be able to expose and motivate the proposed solution of some problems chosen in class either on the blackboard and/or using a computer.

Learning skills:
The acquired knowledge will allow to build the bases for a study related to more specialized aspects of the analysis and approximation of non linear partial differential equations. The student will become familiar with different concepts and techniques related to the topics presented in the course.

Educational objectives

General targets:
acquire basic knowledge of the physical and mathematical aspects of Fluid Mechanics and Kinetic Theory.

Knowledge and understanding:
knowledge of physical principles and modeling assumptions that lead to the equations of fluids and particle systems;
knowledge of fluid and gas equations and their mathematical properties:
weak formulations, existence and uniqueness of solutions,
models for the evolution of singular data.

Applying knowledge and understanding:
the student will be able to
modeling fluid and particle motions, also through the formulation of appropriate
action functionals, discuss the evolution of singularity, use the mathematical tools for
the treatment of fluids and gases in other contexts.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises.

Making judgements:
ability to identify the most significant aspects of the theory,
to know how to evaluate the limits and the advantages of simplifications
operated (incompressibility, absence or presence of viscosity...),
and the limits of mathematical results.

Communication skills:
ability to expose the development of the physical-mathematical
theory for fluids and particle systems, highlighting
the relationship between physical and mathematical aspects;
ability to illustrate the demonstrations,
summarizing the main ideas, and discussing the mathematical details.

Communication skills:
the acquired knowledge will allow a study, individual or given in an LM course, related to numerical aspects
or modeling of fluid mechanics and kinetic theory.

Educational objectives

General skills

The course aims to transmit to students a deep knowledge of the mathematical structure of Quantum Mechanics, of the historical and conceptual path leading to its formulation, and of its relations with other mathematical subjects (as e.g. functional analysis, operator theory, theory of Lie groups and their unitary representations).

Specific skills

A) Knowledge and understanding
After the conclusion of the course, successful students will know and understand the fundamental concepts of Fourier theory, the mathematical analogy between classical mechanics and geometric optics, the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics, and the mathematical structure of Quantum Theory, with a particular emphasis on dynamical aspects (time evolution) and on the analysis of the symmetries of a quantum system (representation of the symmetry group).

B) Applying knowledge and understanding
The general knowledge will be complemented by the application of general concepts to some specific models, and by the ability to analyze symmetries and dynamics of simple quantum systems. Specific simple systems will be analyzed in detail, including the case of a quantum particle in a linear potential, in a harmonic potential, in a uniform magnetic field, and in a Kepler potential (hydrogenoid atom). Successful students will be potentially able to apply the general concepts also to other more complex systems, including non-hydrogenoid atoms, molecules and crystalline solids.

C) Making judgements
The analysis of the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics will make successful students able to autonomously judge the epistemological foundations of a physical theory, and hence to understand its natural range of application and validity. This critical judgement will lead students to privilege an epistemological apophantic approach, with respect to an apodictic one.
Moreover, successful students will be able to autonomously judge the validity of a mathematical statement, through a critical analysis of the hypotheses and of the deductive steps leading to the proof of the statement itself, and to autonomously formulate counterexamples to mathematical statements whenever one of the hypotheses is denied.

D) Communication skills
Successful students will acquire the ability to communicate what has been learned through written themes and oral exams, and to formulate a logically structured speech, with a clear distinction between hypotheses, deduction and thesis.

E) Learning skills
Successful students will acquire the ability to identify the most relevant topics in a subject and to make the logical connections between the topics covered.

Educational objectives

Knowledge and understanding:Successful students will learn various characterizations of Brownianan motion, the fundamental properties of diffusion processes and the main results of stochastic calculus, including the Ito formula.Skills and attributes:Successful students will be able to apply stochastic calculus in various applications, from mathematical finance to physics and biology.

Educational objectives

The course is devoted to the study of multiscale approaches (micro-meso-macro) for multi-agent systems. Typical examples are: vehicular traffic, crowd dynamics, opinion dynamics, flocking/swarming, financial markets and so on.
The course includes lab sessions for the computational part related to the numerica simulation of the models.

1. Knowledge and understanding

Students who have passed the exam will know how to model and study qualitative properties of physical phenomena through several scales of representation: from the microscopic, to the kinetic and the macroscopic one.

2. Applied knowledge and understanding

Students who have passed the exam will be able to use a efficient numerical techniques, deterministic and not, for the simulation of models, and they will be able to code the algorithms in C++ or MATLAB.

3. Making judgments

Students will be able to evaluate the right representation scale of the given phenomenon, the results produced by their programs and to produce tests and simulations.

4. Communication skills

Students will be able to present and explain the modeling choices, the properties of the models, either at the blackboard and/or using a computer.

5. Learning skills

The acquired knowledge will construct the basis to study more research topics related to the modeling of multi-agent systems.

Educational objectives

The course will present the fundamental results related to the analysis and approximation of scalar conservation laws and Hamilton-Jacobi equations. Moreover the course will illustrate a number of models leading to these equations: gas dynamics, traffic models on networks, optimal control problems, image processing, front propagation.

The course includes some Lab sessions to develop programming codes in C++ or MATLAB.

Knowledge and understanding:
Students who have passed the exam will know the main numerical techniques on the topics presented in the course.

Applied knowledge and understanding:
Students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to implement the algorithms in C++ or MATLAB.

Critical and judgmental skills:
Students will be able to evaluate the results produced by their programs and to produce tests and simulations.

Communication skills:
Students will be able to expose and motivate the proposed solution of some problems chosen in class either on the blackboard and/or using a computer.

Learning skills:
The acquired knowledge will allow to build the bases for a study related to more specialized aspects of the analysis and approximation of non linear partial differential equations. The student will become familiar with different concepts and techniques related to the topics presented in the course.

Educational objectives

General objectives:

Many models in mathematical physics and natural sciences in general have variational principles (the principle of minimal energy, minimal action, ...) which describe their equilibrium configurations and dynamic evolutions.
The aim of the course is to make students aware of the variety of problems that can be addressed with variational techniques and to provide them with the basic tools and mathematical language for analyzing the models arising in natural sciences.

Specific objectives:

Knowledge and understanding:

at the end of the course the student will have acquired the basic notions and results on the direct method of calculus variations, conditions for semicontinuity, asymptotic analysis via Gamma convergence, and she/he will be able to apply this methods in various contexts about which they will be provided the functional bases at least in dimension 1 (integral functionals and Sobolev spaces, geometric functionals and elements of geometric measure theory).

Apply knowledge and understanding:

at the end of the course the student will be able to begin the study of advanced calculus of variations. She/he will also be able to formulate a simple variational model (for example linked to a specific application) and analyze its asymptotic behavior or identify the characteristics that make it a robust model.

Critical and judgmental skills:

The student will have the basics to connect and use tools covered in various moments of his studies ranging from analysis, mathematical physics and to probability. She/he will therefore be able to appreciate the interest of a mathematical question in relation also to its use to answer a question coming from an applied problem.

Communication skills:

ability to rigorously expose the theoretical contents of the course and also ability to formulate the problem under consideration by understanding the role of deriving the right model and its analysis. Ability to explain moreover the results in the language related to the application under consideration, potentially understandable by non expert in calculating variations.

Learning ability:

the acquired knowledge will allow to face a possible master's thesis work in the field of applied mathematics in natural sciences both with a more theoretical approach and in connection with the analysis of a specific model of interest for applications.

Educational objectives

Knowledge and understanding:The course gives to successful students some advanced tools for the study of various linear and nonlinear PDE's. They will reach a good familiarity with the most recent notions of solutions and their qualitative properties.Skills and attributes:Successful students will able to deal with the advanced study of the solutions to various types of linear and nonlinear PDE's.

Educational objectives

1) Knowledge and understanding
At the end of the course, the students will know and understand:
a) the idea of control system and of differential inclusion, and their basic properties;
b) thr idea of optimal control and necessary and/or sufficient conditions for its existence;
c) the relationship between optimal solutions of a control problem and the Hamilton-Jacobi-Bellman equation;
d) the idea of viscosity solution for the Hamilton-Jacobi equation.

2) Applying knowledge and understanding
At the end of the course, the students will be able to:
a) write the mathematical formulation of an optimal control problem;
b) determine, using the Pontryagin Maximum Principle, the optimal solutions of an optimal control problem;
c) analyze, from a theoretical point of view, the solutions of an optimal control problem through the study of the associated Hamilton-Jacobi-Bellman equation.

3) Making judgements
During the lessons, several problems will be proposed to the students.
Thanks to the autonomous resolution of the problems, and the subsequent discussion in the classroom, the students will acquire both the ability to evaluate their knowledge and the ability to tackle a wide range of optimal control problems.

4) Communication skills
The written form of the exercises, assigned either during lessons or during the written test, and the oral exam will allow the students to evaluate their skill in correctly communicating the knowledges acquired during the course.

5) Learning skills
At the end of the course the students will be able to analyze optimal control problems; such skill is acquired by means of several model problems assigned during the course.

Educational objectives

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

Educational objectives

General objectives:

Many models in mathematical physics and natural sciences in general have variational principles (the principle of minimal energy, minimal action, ...) which describe their equilibrium configurations and dynamic evolutions.
The aim of the course is to make students aware of the variety of problems that can be addressed with variational techniques and to provide them with the basic tools and mathematical language for analyzing the models arising in natural sciences.

Specific objectives:

Knowledge and understanding:

at the end of the course the student will have acquired the basic notions and results on the direct method of calculus variations, conditions for semicontinuity, asymptotic analysis via Gamma convergence, and she/he will be able to apply this methods in various contexts about which they will be provided the functional bases at least in dimension 1 (integral functionals and Sobolev spaces, geometric functionals and elements of geometric measure theory).

Apply knowledge and understanding:

at the end of the course the student will be able to begin the study of advanced calculus of variations. She/he will also be able to formulate a simple variational model (for example linked to a specific application) and analyze its asymptotic behavior or identify the characteristics that make it a robust model.

Critical and judgmental skills:

The student will have the basics to connect and use tools covered in various moments of his studies ranging from analysis, mathematical physics and to probability. She/he will therefore be able to appreciate the interest of a mathematical question in relation also to its use to answer a question coming from an applied problem.

Communication skills:

ability to rigorously expose the theoretical contents of the course and also ability to formulate the problem under consideration by understanding the role of deriving the right model and its analysis. Ability to explain moreover the results in the language related to the application under consideration, potentially understandable by non expert in calculating variations.

Learning ability:

the acquired knowledge will allow to face a possible master's thesis work in the field of applied mathematics in natural sciences both with a more theoretical approach and in connection with the analysis of a specific model of interest for applications.

Educational objectives

Knowledge and understanding:The course gives to successful students some advanced tools for the study of various linear and nonlinear PDE's. They will reach a good familiarity with the most recent notions of solutions and their qualitative properties.Skills and attributes:Successful students will able to deal with the advanced study of the solutions to various types of linear and nonlinear PDE's.

Educational objectives

General objectives: to acquire basic knowledge in stochastic process theory and in stochastic modeling
of real phenomena.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results concerning stochastic processes in discrete and continuous time, on discrete structures such as graphs or on continuous spaces.

Apply knowledge and understanding: at the end of the course the student will be able to model the temporal evolution of various real phenomena through stochastic processes, to analyze the stationarity and / or temporal reversibility of stochastic processes, to calculate probabilities of absorption and expected absorption times, to simulate stochastic processes and to estimate the rate of convergence at equilibrium.

Critical and judgmental skills: the student will have the basis to study stochastic dynamic systems and acquire the ability to evaluate the goodness of a model compared to others in the modeling of real phenomena.

Communication skills: having to take an oral theory test, students will develop the communication skills necessary to expose the mathematical theory and the various models considered in the course.

Learning skills: the acquired knowledge will allow a more in-depth study of stochastic processes both on discrete and continuous spaces, helping the student to study other courses such as stochastic calculus.

Educational objectives

General targets: To acquire advanced knowledge in the theory of dynamical systems.

Specific targets:

Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on applications in mechanics and applied sciences in general. They will learn elements of stability theory and hyperbolic theory (such as homoclinic intersections and existence of chaotic motions). They will also learn elements of the theory of topological dynamical systems and of ergodic theory.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study stability problems of equilibria and cycles, both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.

Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.

Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.

Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.

Educational objectives

The course is devoted to the study of multiscale approaches (micro-meso-macro) for multi-agent systems. Typical examples are: vehicular traffic, crowd dynamics, opinion dynamics, flocking/swarming, financial markets and so on.
The course includes lab sessions for the computational part related to the numerica simulation of the models.

1. Knowledge and understanding

Students who have passed the exam will know how to model and study qualitative properties of physical phenomena through several scales of representation: from the microscopic, to the kinetic and the macroscopic one.

2. Applied knowledge and understanding

Students who have passed the exam will be able to use a efficient numerical techniques, deterministic and not, for the simulation of models, and they will be able to code the algorithms in C++ or MATLAB.

3. Making judgments

Students will be able to evaluate the right representation scale of the given phenomenon, the results produced by their programs and to produce tests and simulations.

4. Communication skills

Students will be able to present and explain the modeling choices, the properties of the models, either at the blackboard and/or using a computer.

5. Learning skills

The acquired knowledge will construct the basis to study more research topics related to the modeling of multi-agent systems.

Educational objectives

General targets:
to acquire basic and advanced knowledge and hands-on experience on
some classic topics in finite-dimensional optimization.

Specific targets

Knowledge and understanding:
Understanding of the theoretical foundations of optimization theory and of the main
algorithm classes for the solution of optimization problems.

Applying knowledge and understanding:
the student will be able to identify relevant characteristics of optimization problems
and to select the most appropriate solution method for a given problem, also taking into account practical constraints due to the applicative environment (for example, the required accuracy or time limits). In addition the student will be able to correctly
analyze the results provided by commercial or ad-hoc resolution software.

Making judgements:
ability to enucleate the most significant aspects of an optimization problem and of its solution methods.

Communication skills:
ability to enucleate the significant points of the theory,
to know how to illustrate the most interesting parts with appropriate examples,
to discuss mathematically the most subtle points.

Learning skills:
the acquired knowledge will allow the student to undertake
more advanced studies in optimization and to be able to work
in industrial and research environments where optimization is used.

Educational objectives

1) Knowledge and understanding
At the end of the course, the students will know and understand:
a) the idea of control system and of differential inclusion, and their basic properties;
b) thr idea of optimal control and necessary and/or sufficient conditions for its existence;
c) the relationship between optimal solutions of a control problem and the Hamilton-Jacobi-Bellman equation;
d) the idea of viscosity solution for the Hamilton-Jacobi equation.

2) Applying knowledge and understanding
At the end of the course, the students will be able to:
a) write the mathematical formulation of an optimal control problem;
b) determine, using the Pontryagin Maximum Principle, the optimal solutions of an optimal control problem;
c) analyze, from a theoretical point of view, the solutions of an optimal control problem through the study of the associated Hamilton-Jacobi-Bellman equation.

3) Making judgements
During the lessons, several problems will be proposed to the students.
Thanks to the autonomous resolution of the problems, and the subsequent discussion in the classroom, the students will acquire both the ability to evaluate their knowledge and the ability to tackle a wide range of optimal control problems.

4) Communication skills
The written form of the exercises, assigned either during lessons or during the written test, and the oral exam will allow the students to evaluate their skill in correctly communicating the knowledges acquired during the course.

5) Learning skills
At the end of the course the students will be able to analyze optimal control problems; such skill is acquired by means of several model problems assigned during the course.

Educational objectives

The course will present the fundamental results related to the analysis and approximation of scalar conservation laws and Hamilton-Jacobi equations. Moreover the course will illustrate a number of models leading to these equations: gas dynamics, traffic models on networks, optimal control problems, image processing, front propagation.

The course includes some Lab sessions to develop programming codes in C++ or MATLAB.

Knowledge and understanding:
Students who have passed the exam will know the main numerical techniques on the topics presented in the course.

Applied knowledge and understanding:
Students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to implement the algorithms in C++ or MATLAB.

Critical and judgmental skills:
Students will be able to evaluate the results produced by their programs and to produce tests and simulations.

Communication skills:
Students will be able to expose and motivate the proposed solution of some problems chosen in class either on the blackboard and/or using a computer.

Learning skills:
The acquired knowledge will allow to build the bases for a study related to more specialized aspects of the analysis and approximation of non linear partial differential equations. The student will become familiar with different concepts and techniques related to the topics presented in the course.

Educational objectives

General targets:
acquire basic knowledge of the physical and mathematical aspects of Fluid Mechanics and Kinetic Theory.

Knowledge and understanding:
knowledge of physical principles and modeling assumptions that lead to the equations of fluids and particle systems;
knowledge of fluid and gas equations and their mathematical properties:
weak formulations, existence and uniqueness of solutions,
models for the evolution of singular data.

Applying knowledge and understanding:
the student will be able to
modeling fluid and particle motions, also through the formulation of appropriate
action functionals, discuss the evolution of singularity, use the mathematical tools for
the treatment of fluids and gases in other contexts.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises.

Making judgements:
ability to identify the most significant aspects of the theory,
to know how to evaluate the limits and the advantages of simplifications
operated (incompressibility, absence or presence of viscosity...),
and the limits of mathematical results.

Communication skills:
ability to expose the development of the physical-mathematical
theory for fluids and particle systems, highlighting
the relationship between physical and mathematical aspects;
ability to illustrate the demonstrations,
summarizing the main ideas, and discussing the mathematical details.

Communication skills:
the acquired knowledge will allow a study, individual or given in an LM course, related to numerical aspects
or modeling of fluid mechanics and kinetic theory.

Educational objectives

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

Educational objectives

Knowledge and understanding:Successful students will learn various characterizations of Brownianan motion, the fundamental properties of diffusion processes and the main results of stochastic calculus, including the Ito formula.Skills and attributes:Successful students will be able to apply stochastic calculus in various applications, from mathematical finance to physics and biology.

Educational objectives

General objectives: to acquire basic knowledge in stochastic process theory and in stochastic modeling
of real phenomena.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results concerning stochastic processes in discrete and continuous time, on discrete structures such as graphs or on continuous spaces.

Apply knowledge and understanding: at the end of the course the student will be able to model the temporal evolution of various real phenomena through stochastic processes, to analyze the stationarity and / or temporal reversibility of stochastic processes, to calculate probabilities of absorption and expected absorption times, to simulate stochastic processes and to estimate the rate of convergence at equilibrium.

Critical and judgmental skills: the student will have the basis to study stochastic dynamic systems and acquire the ability to evaluate the goodness of a model compared to others in the modeling of real phenomena.

Communication skills: having to take an oral theory test, students will develop the communication skills necessary to expose the mathematical theory and the various models considered in the course.

Learning skills: the acquired knowledge will allow a more in-depth study of stochastic processes both on discrete and continuous spaces, helping the student to study other courses such as stochastic calculus.

Educational objectives

General targets: To acquire advanced knowledge in the theory of dynamical systems.

Specific targets:

Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on applications in mechanics and applied sciences in general. They will learn elements of stability theory and hyperbolic theory (such as homoclinic intersections and existence of chaotic motions). They will also learn elements of the theory of topological dynamical systems and of ergodic theory.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study stability problems of equilibria and cycles, both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.

Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.

Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.

Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.

Educational objectives

General targets:
acquire basic knowledge of the physical and mathematical aspects of Fluid Mechanics and Kinetic Theory.

Knowledge and understanding:
knowledge of physical principles and modeling assumptions that lead to the equations of fluids and particle systems;
knowledge of fluid and gas equations and their mathematical properties:
weak formulations, existence and uniqueness of solutions,
models for the evolution of singular data.

Applying knowledge and understanding:
the student will be able to
modeling fluid and particle motions, also through the formulation of appropriate
action functionals, discuss the evolution of singularity, use the mathematical tools for
the treatment of fluids and gases in other contexts.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises.

Making judgements:
ability to identify the most significant aspects of the theory,
to know how to evaluate the limits and the advantages of simplifications
operated (incompressibility, absence or presence of viscosity...),
and the limits of mathematical results.

Communication skills:
ability to expose the development of the physical-mathematical
theory for fluids and particle systems, highlighting
the relationship between physical and mathematical aspects;
ability to illustrate the demonstrations,
summarizing the main ideas, and discussing the mathematical details.

Communication skills:
the acquired knowledge will allow a study, individual or given in an LM course, related to numerical aspects
or modeling of fluid mechanics and kinetic theory.

Educational objectives

Knowledge and understanding:Successful students will learn various characterizations of Brownianan motion, the fundamental properties of diffusion processes and the main results of stochastic calculus, including the Ito formula.Skills and attributes:Successful students will be able to apply stochastic calculus in various applications, from mathematical finance to physics and biology.

Educational objectives

General objectives: to acquire the basic knowledge and techniques of the combinatorics of permutations, enumerative combinatorics, combinatorics of integer partitions, generating functions and understand its main applications.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to Combinatorics of permutations (with particular regard to enumerations, representation with trees, cycles, linear orderings, random generation) and enumerative combinatorics (especially concerning its algebraic aspects, via generating functions). She will also know at least the set of the most significant problems in which these theories find applications.

Apply knowledge and understanding: the student will be able to solve algebraic-combinatorial problems requiring the use of techniques related to the theories of combinatorics of permutations, enumerative combinatorics, of posets and integer partitions, and to discuss how problems (in non-purely mathematical environments) can be modeled by means of the acquired tools.

Critical and judgmental skills: the student will have the basis to analyze how the topics of combinatorics and Algebra and Linear Algebra treated in basic courses can find applications in different fields and be an essential tool in solving concrete problems.

Communication skills: The learner will have the ability to communicate rigorously the ideas and contents shown in the course.

Learning skills: the acquired knowledge will allow the student to carry on an autonomous study in a possible interdisciplinary context (for those who have knowledge and interests in Applied Mathematics, Genetics, Computer Science, Data Science).

Educational objectives

General objectives: to acquire the basic knowledge and techniques of the Theory of graphs and Hypergraphs, in its algebraic and extremal aspects, and on the theory of algebraic codes.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the Theory of graphs and Hypergraphs, in its algebraic and extremal aspects, and on the theory of algebraic codes (with particular regard to algorithms on graphs, representation with trees, cycles, linear orderings, random generation) and Ramsey Theory. She will also know at least the set of the most significant problems in which these theories find applications.

Apply knowledge and understanding: the student will be able to solve algebraic-combinatorial problems in graphs theory and Theory of algebraic codes, requiring the use of related techniques, and to discuss how problems (in non-purely mathematical environments) can be modeled by means of the acquired tools.

Critical and judgmental skills: the student will have the basis to analyze how the topics of Graph and coding theory an find applications in different fields and be an essential tool in solving concrete problems.

Communication skills: The learner will have the ability to communicate rigorously the ideas and contents shown in the course.

Learning skills: the acquired knowledge will allow the student to carry on an autonomous study in a possible interdisciplinary context (for those who have knowledge and interests in Applied Mathematics, Genetics, Computer Science, Data Science).

Educational objectives

General skills
This course is designed to explore the fundamentals of time series analysis.

Specific skills

Knowledge and understanding:
Knowing and understanding of basic results of mathematical models of time series: stationary and non stationary processes, multivariate linear models, ARIMA models, spectral analysis, trend, test of serial independance.

Applying knowledge and understanding:
Be able to analyze simple data series, to estimate parameters, to extract trend and noise, to perform residual diagnostics.

Making judgements:
Be able to understand relationship to basic linear algebra, analysis, probability and statistics.

Communication skills:
Be able to communicate what has been learned during the laboratory and oral exam.

Learning skills:
Be able to learning the specific terminology and advanced methods on time series.

Educational objectives

1. Knowledge and understanding
Students who have passed the exam will know and understand the main tools for Data Analysis: Gaussian models, linear models, principal component analysis, factor analysis, discriminant analysis, analysis of
canonical correlation, multi-dimensional scaling, causal models, Markov chains, random graphs, graph-based algorithms.

2. Applied knowledge and understanding
Students who pass the exam will be able to solve Data Mining problems, including model selection, prediction, classification, clustering, dimension reduction, feature extraction, causal inference.

3. Making judgments
Students will be able to evaluate the results produced by their programs and to produce tests and simulations.

4. Communication skills
Students will be able to present and explain the solution of some problems and excercises either at the blackboard and/or using a computer.

5. Learning skills
The acquired knowledge will construct the basis to study more specialized topics of Data Science and the numerical methods in this area.

Educational objectives

General objectives

Acquiring basic knowledge on the mathematical methods used in artificial intelligence modeling, with particular attention to "machine learning".

Specific objectives

Knowledge and understanding: at the end of the course the student will have knowledge of the basic notions and results (mainly in the areas of stochastic processes and statistical mechanics) used in the study of the main models of neural networks (e.g., Hopfield networks, Boltzmann machines, feed-forward networks).

Apply knowledge and understanding: the student will be able to identify the optimal architecture for a certain task and to solve the resulting model by determining a phase diagram; the student will have the basis to independently develop algorithms for learning and retrieval.

Critical and judgmental skills: the student will be able to determine the parameters that control the qualitative behaviour of a neural network and to estimate the values of these parameters that allow a good performance of the network; she/he will also be able to investigate the analogies and relationships between the topics covered during the course and during courses dedicated to statistics and data analysis.

Communication skills: ability to expose the contents in the oral and written part of the verification, possibly by means of presentations.

Learning skills: the knowledge acquired will allow a study, individual or taught in a LM course, related to more specialised aspects of statistical mechanics, development of algorithms, usage of big data.

Educational objectives

General objectives: to acquire knowledge in High dimensional Probability and Statistics with applications to Data Science

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions of High Dimensional Probability and Statistics and will be familiar with algorithms used to solve some relevant problems in Data Science.

Apply knowledge and understanding: at the end of the course the student will be able to solve some problems concerning high dimensional random geometric structures, data dimension reduction, statistical learning and high dimensional regression

Critical and judgmental skills: the student will realize the ideas behind several algorithms and software used in Data Science,
understand optimal conditions and/or possible limits for applications

Communication skills: the student must show the ability to present the contents of the course in the oral part of the assessment and in the solution of problems in the written test.

Learning skills: the acquired knowledge will allow a multidisciplinary understanding of several problems motivated by data science and will facilitate the study into some very active research fields.

Educational objectives

General Goals
The course deals with some fundamental issues of contemporary research in algorithms in the field of computational complexity, probabilistic algorithms and machine learning.

Educational objectives

General Objectives:
The goal of the course is to hand down the foundations of cryptography, which is at the heart of security in nowadays digital applications.

Specific Objectives:
The students will learn the methodology of provable security, which allows to prove security of modern cryptosystems in a mathematically sound way.

Knowledge and Understanding:
-) Knowledge of the mathematical foundations of modern cryptography.
-) Knowledge of the main hardness assumptions, on which the security of cryptographic constructions is based.
-) Knowledge of the cryptographic schemes currently used in real life. Understanding of their (practical and theoretical) properties.

Applying knowledge and understanding:
-) How to select the right cryptographic scheme for a given application.
-) How to analyze the security of a given cryptographic scheme.

Critiquing and judgmental skills:
The students will be able to judge whether a given cryptographic scheme is secure or not.

Communication Skills:
How to describe the security of a cryptographic construction in the language of provable security.

Ability of learning:
The students interested in research will learn what are the main open challenges in the area, and will obtain the necessary background for a deeper study of the subjects.

Educational objectives

General skills
This course is designed to explore the fundamentals of time series analysis.

Specific skills

Knowledge and understanding:
Knowing and understanding of basic results of mathematical models of time series: stationary and non stationary processes, multivariate linear models, ARIMA models, spectral analysis, trend, test of serial independance.

Applying knowledge and understanding:
Be able to analyze simple data series, to estimate parameters, to extract trend and noise, to perform residual diagnostics.

Making judgements:
Be able to understand relationship to basic linear algebra, analysis, probability and statistics.

Communication skills:
Be able to communicate what has been learned during the laboratory and oral exam.

Learning skills:
Be able to learning the specific terminology and advanced methods on time series.

Educational objectives

1. Knowledge and understanding
Students who have passed the exam will know and understand the main tools for Data Analysis: Gaussian models, linear models, principal component analysis, factor analysis, discriminant analysis, analysis of
canonical correlation, multi-dimensional scaling, causal models, Markov chains, random graphs, graph-based algorithms.

2. Applied knowledge and understanding
Students who pass the exam will be able to solve Data Mining problems, including model selection, prediction, classification, clustering, dimension reduction, feature extraction, causal inference.

3. Making judgments
Students will be able to evaluate the results produced by their programs and to produce tests and simulations.

4. Communication skills
Students will be able to present and explain the solution of some problems and excercises either at the blackboard and/or using a computer.

5. Learning skills
The acquired knowledge will construct the basis to study more specialized topics of Data Science and the numerical methods in this area.

Educational objectives

General objectives: to acquire the basic knowledge and techniques of the combinatorics of permutations, enumerative combinatorics, combinatorics of integer partitions, generating functions and understand its main applications.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to Combinatorics of permutations (with particular regard to enumerations, representation with trees, cycles, linear orderings, random generation) and enumerative combinatorics (especially concerning its algebraic aspects, via generating functions). She will also know at least the set of the most significant problems in which these theories find applications.

Apply knowledge and understanding: the student will be able to solve algebraic-combinatorial problems requiring the use of techniques related to the theories of combinatorics of permutations, enumerative combinatorics, of posets and integer partitions, and to discuss how problems (in non-purely mathematical environments) can be modeled by means of the acquired tools.

Critical and judgmental skills: the student will have the basis to analyze how the topics of combinatorics and Algebra and Linear Algebra treated in basic courses can find applications in different fields and be an essential tool in solving concrete problems.

Communication skills: The learner will have the ability to communicate rigorously the ideas and contents shown in the course.

Learning skills: the acquired knowledge will allow the student to carry on an autonomous study in a possible interdisciplinary context (for those who have knowledge and interests in Applied Mathematics, Genetics, Computer Science, Data Science).

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.

Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.

Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.

Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.

Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.

Educational objectives

General objectives: to acquire basic knowledge in stochastic process theory and in stochastic modeling
of real phenomena.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results concerning stochastic processes in discrete and continuous time, on discrete structures such as graphs or on continuous spaces.

Apply knowledge and understanding: at the end of the course the student will be able to model the temporal evolution of various real phenomena through stochastic processes, to analyze the stationarity and / or temporal reversibility of stochastic processes, to calculate probabilities of absorption and expected absorption times, to simulate stochastic processes and to estimate the rate of convergence at equilibrium.

Critical and judgmental skills: the student will have the basis to study stochastic dynamic systems and acquire the ability to evaluate the goodness of a model compared to others in the modeling of real phenomena.

Communication skills: having to take an oral theory test, students will develop the communication skills necessary to expose the mathematical theory and the various models considered in the course.

Learning skills: the acquired knowledge will allow a more in-depth study of stochastic processes both on discrete and continuous spaces, helping the student to study other courses such as stochastic calculus.

Educational objectives

General Goals
The course deals with some fundamental issues of contemporary research in algorithms in the field of computational complexity, probabilistic algorithms and machine learning.

Educational objectives

The course is devoted to the study of multiscale approaches (micro-meso-macro) for multi-agent systems. Typical examples are: vehicular traffic, crowd dynamics, opinion dynamics, flocking/swarming, financial markets and so on.
The course includes lab sessions for the computational part related to the numerica simulation of the models.

1. Knowledge and understanding

Students who have passed the exam will know how to model and study qualitative properties of physical phenomena through several scales of representation: from the microscopic, to the kinetic and the macroscopic one.

2. Applied knowledge and understanding

Students who have passed the exam will be able to use a efficient numerical techniques, deterministic and not, for the simulation of models, and they will be able to code the algorithms in C++ or MATLAB.

3. Making judgments

Students will be able to evaluate the right representation scale of the given phenomenon, the results produced by their programs and to produce tests and simulations.

4. Communication skills

Students will be able to present and explain the modeling choices, the properties of the models, either at the blackboard and/or using a computer.

5. Learning skills

The acquired knowledge will construct the basis to study more research topics related to the modeling of multi-agent systems.

Educational objectives

General Objectives:
The goal of the course is to hand down the foundations of cryptography, which is at the heart of security in nowadays digital applications.

Specific Objectives:
The students will learn the methodology of provable security, which allows to prove security of modern cryptosystems in a mathematically sound way.

Knowledge and Understanding:
-) Knowledge of the mathematical foundations of modern cryptography.
-) Knowledge of the main hardness assumptions, on which the security of cryptographic constructions is based.
-) Knowledge of the cryptographic schemes currently used in real life. Understanding of their (practical and theoretical) properties.

Applying knowledge and understanding:
-) How to select the right cryptographic scheme for a given application.
-) How to analyze the security of a given cryptographic scheme.

Critiquing and judgmental skills:
The students will be able to judge whether a given cryptographic scheme is secure or not.

Communication Skills:
How to describe the security of a cryptographic construction in the language of provable security.

Ability of learning:
The students interested in research will learn what are the main open challenges in the area, and will obtain the necessary background for a deeper study of the subjects.

Educational objectives

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

Educational objectives

General objectives: to acquire the basic knowledge and techniques of the Theory of graphs and Hypergraphs, in its algebraic and extremal aspects, and on the theory of algebraic codes.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the Theory of graphs and Hypergraphs, in its algebraic and extremal aspects, and on the theory of algebraic codes (with particular regard to algorithms on graphs, representation with trees, cycles, linear orderings, random generation) and Ramsey Theory. She will also know at least the set of the most significant problems in which these theories find applications.

Apply knowledge and understanding: the student will be able to solve algebraic-combinatorial problems in graphs theory and Theory of algebraic codes, requiring the use of related techniques, and to discuss how problems (in non-purely mathematical environments) can be modeled by means of the acquired tools.

Critical and judgmental skills: the student will have the basis to analyze how the topics of Graph and coding theory an find applications in different fields and be an essential tool in solving concrete problems.

Communication skills: The learner will have the ability to communicate rigorously the ideas and contents shown in the course.

Learning skills: the acquired knowledge will allow the student to carry on an autonomous study in a possible interdisciplinary context (for those who have knowledge and interests in Applied Mathematics, Genetics, Computer Science, Data Science).

Educational objectives

General objectives

Acquiring basic knowledge on the mathematical methods used in artificial intelligence modeling, with particular attention to "machine learning".

Specific objectives

Knowledge and understanding: at the end of the course the student will have knowledge of the basic notions and results (mainly in the areas of stochastic processes and statistical mechanics) used in the study of the main models of neural networks (e.g., Hopfield networks, Boltzmann machines, feed-forward networks).

Apply knowledge and understanding: the student will be able to identify the optimal architecture for a certain task and to solve the resulting model by determining a phase diagram; the student will have the basis to independently develop algorithms for learning and retrieval.

Critical and judgmental skills: the student will be able to determine the parameters that control the qualitative behaviour of a neural network and to estimate the values of these parameters that allow a good performance of the network; she/he will also be able to investigate the analogies and relationships between the topics covered during the course and during courses dedicated to statistics and data analysis.

Communication skills: ability to expose the contents in the oral and written part of the verification, possibly by means of presentations.

Learning skills: the knowledge acquired will allow a study, individual or taught in a LM course, related to more specialised aspects of statistical mechanics, development of algorithms, usage of big data.

Educational objectives

General objectives: to acquire knowledge in High dimensional Probability and Statistics with applications to Data Science

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions of High Dimensional Probability and Statistics and will be familiar with algorithms used to solve some relevant problems in Data Science.

Apply knowledge and understanding: at the end of the course the student will be able to solve some problems concerning high dimensional random geometric structures, data dimension reduction, statistical learning and high dimensional regression

Critical and judgmental skills: the student will realize the ideas behind several algorithms and software used in Data Science,
understand optimal conditions and/or possible limits for applications

Communication skills: the student must show the ability to present the contents of the course in the oral part of the assessment and in the solution of problems in the written test.

Learning skills: the acquired knowledge will allow a multidisciplinary understanding of several problems motivated by data science and will facilitate the study into some very active research fields.