Mathematics

Educational objectives

Knowledge and ability to understand: The course aims to provide useful knowledge to understand some aspects of theoretical physics, and specifically of quantum mechanics and statistical mechanics. Particular attention will be dedicated
to the analysis of the hypotheses founding the two disciplines. Through the study of these issues the student will be able to understand the evolution of physics that took place at the beginning of the last century, the impact they had on the development of society and their current importance.

Ability to apply knowledge and understanding: The course is aimed at providing tools for the analysis and evaluation of physical phenomena on atomic scales and on the collective behaviors of large numbers of interacting particles. This knowledge can also be exported to fields other than those proposed in the course.

Making judgments: Through the study of the theoretical approaches to quantum mechanics and statistics, the student will be able to improve his ability to interpret the real.

Communication skills: The development of communication skills, mainly oral, will be stimulated through the discussion in the classroom and possibly with participation in seminar activities.

Learning ability: The ability to learn will be stimulated through the discussion in the classroom, which will include interactive aspects also aimed at verifying the effective understanding of the topics covered. The learning capacity will also be stimulated by integrative teaching supports (original articles) in order to develop the application skills.

Educational objectives

General targets: acquisition of basic knowledge in automata theory.

Specific targets:

Knowledge and understanding: at the end of the course, students will be acquainted with the notions of deterministic and complete automaton, recognizable language, non-deterministic automaton, and rational language, together with theorems describing some fundamental properties, of algebraic and combinatorial nature, of such structures (description of languages accepted by finite automata in term of finite index congruences, rational operations in the free semigroup of strings, non deterministic models and minimal automata).

Apply Knowledge and understanding: at the end of the course, students will be able to solve simple problems of automata theory, by using algebraic and combinatorial techniques: construction of automata for the acceptance of languages, decidability and algorithmic properties of automata, tools to verify the non-recognazibility of formal languages.

Analytical and judgment abilities: successful students will be able to manipulate the basic objects of the theory and they will be able to understand the proofs of some theorems that are relevant in the theory of automata. Moreover they will be able to analyse relations with topics of mathematical theory of formal languages and theory of codes.

Communication skills: the student will be able to present, in a written classwork, his knowledge of the theory and the solutions of the exercises.

Learning skills: the acquired knowledge and skills will permit the student to study, at individual level or in a course taught in the LM, more advanced aspects of automata theory and of mathematical theory of formal languages.

Educational objectives

General Goals: rigorous knowledge of probabilistic models from
applications to the relationship with other part of mathematic.

Specific goals:

Knowledge and understanding: at the end of the course the student will
have acquired the basic notions and results related to probability
spaces, random variables, independence, laws of large numbers,
characteristic functions, weak convergence, limit theorems.

Apply knowledge and understanding: at the end of the
course the student will be able to solve simple problems that require
the use of probabilistic techniques both in applications and in
problems of pure mathematics.

Critical and judgmental skills: the student will have the basis to
analyze the analogies and the relationships between the topics covered
and topics of analysis or mathematical physics; will also acquire the
tools that have historically led to the solution of classical
problems.

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 study, individual
or given in an LM course, related to more specialized aspects of
probability.

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 targets:
acquire basic specialist knowledge on
some classic topics of Physics-Mathematics.

Knowledge and understanding:
knowledge of the theory of compact self-adjoint operators, of the applications of this theory
to the theory of potential; basic knowledge of Hamiltonian Mechanics
and of Quantum Mechanics.

Applying knowledge and understanding:
the student will be able to
analyze the spectrum of operators, also for unbounbed operators; to determine
the eigenvalues of the Laplacian in domains with symmetries;
translate into Hamiltonian formalism
the Lagrangian problems and solve them for quadatrure;
discuss the solution of the
Schroedinger equation in simple but physically significant cases.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises, subject to written verification.

Making judgements:
ability to enucleate the most significant aspects of the potential theory
and of the theory of motion,
ability to reflect on similarities and differences between the classical case
and the quantum one.

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 face
the mathematical-physcs courses on more specialized subjects,
and will allow the student to understand, even independently,
the physical relevance of mathematical questions discussed in other courses.

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

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

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.

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:
Successful students will be able to deal with topics concerning Functional Analysis and foundations of the theory of Operator Algebras, will become proficient and acquainted with subjects such as Spectral Theory, closed, hermitian and selfadjoint operators, C* Algebras and von Neumann Algebras.

Skills and attributes:
Successful students will be able to study the structure of linear operators like simple perturbations of the Laplacian and to use the theory of Operator Algebras in many fundamental questions, from Harmonic analysis to Quantum Mechanics.

Educational objectives

General goals.
To acquire basic notions and skills in algebraic geometry.

Specific goals.

Knowledge and comprehension: at the end of the course the student will have learnt basic notions and results on affine and projective varieties, quasi-projective varieties, dimension, and first local/global properties.

Applied knowledge and comprehension: at the end of the course the student will the skill to go into and appreciate modern algebraic geometry.

Critical development: the student will appreciate the interaction between different fields such as commutative
algebra, complex analysis, analytic and projective geometry.

Communication skills: the ability to make a clear presentation of parts of the theory
introduced in the course.

Learning skills: the knowledge acquired will be useful in studying more specialized courses in algebraic and complex geometry.

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 bases 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

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

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 hyperbolic systems and applications in mechanics, like stability theory. Moreover, they will learn part of the general theory of hyperbolic invariant sets, with applications to homoclinic intersections, chaotic motion, and ergodic theory, in the framework of concrete mechanical systems.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study equilibrium stability problems 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 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:
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 aims of the course: acquire basic notions of algebraic topology.

Detailed aims of the course:

Notions and comprehension: in this course the student will acquire notions and basic results about homology and cohomology and about the theory of characteristic classes of vector bundles.

Applying acquired notions: in this course the student will learn how to solve simple exercises using techniques from homotopy theory, homology and cohomology and algebraic structures related to them; such ability will be verified in the written test.

Critical thinking abilities: in this course the student will discover analogies between the treated topics and other topics in topology (seen in Geometria 1-2), differential geometry (seen in Geometria 2, Geometira differenziale, Istituzioni di geometria superiore). Moreover, the student will learn tools that lead to the solution of some classical problems.

Communication abilities: the student will learn how to comunicate the acquired mathematical content; such abilities will be verified during the oral examination and possibly in some of the theoretical questions in the written test.

Learning abilities: the acquired knowledge will allow the student to pursue a deeper investigation of the topology of differentiable manifolds and algebraic varieties.

Educational objectives

General objectives: to acquire specialized knowledge about representation theory of Lie algebras.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired advanced notions and results related to representation theory of finite dimensional Lie algebras and an introduction to homological methods in algebra.

Apply knowledge and understanding: at the end of the course the student will be able to read research articles related to the topics of the course.

Critical and judgment skills: the student will have the basics to analyze the analogies and relationships between the topics covered and their developments in representation theory and
homological algebra.

Communication skills: ability to present topics in seminars using techniques and results addressed in the course.

Learning skills: the knowledge acquired will prepare the students to start a research activity in algebra and geometry.

Educational objectives

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 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

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 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

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 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 ability to understand: The course aims to provide useful knowledge to understand some aspects of theoretical physics, and specifically of quantum mechanics and statistical mechanics. Particular attention will be dedicated
to the analysis of the hypotheses founding the two disciplines. Through the study of these issues the student will be able to understand the evolution of physics that took place at the beginning of the last century, the impact they had on the development of society and their current importance.

Ability to apply knowledge and understanding: The course is aimed at providing tools for the analysis and evaluation of physical phenomena on atomic scales and on the collective behaviors of large numbers of interacting particles. This knowledge can also be exported to fields other than those proposed in the course.

Making judgments: Through the study of the theoretical approaches to quantum mechanics and statistics, the student will be able to improve his ability to interpret the real.

Communication skills: The development of communication skills, mainly oral, will be stimulated through the discussion in the classroom and possibly with participation in seminar activities.

Learning ability: The ability to learn will be stimulated through the discussion in the classroom, which will include interactive aspects also aimed at verifying the effective understanding of the topics covered. The learning capacity will also be stimulated by integrative teaching supports (original articles) in order to develop the application skills.

Educational objectives

General targets: acquisition of basic knowledge in automata theory.

Specific targets:

Knowledge and understanding: at the end of the course, students will be acquainted with the notions of deterministic and complete automaton, recognizable language, non-deterministic automaton, and rational language, together with theorems describing some fundamental properties, of algebraic and combinatorial nature, of such structures (description of languages accepted by finite automata in term of finite index congruences, rational operations in the free semigroup of strings, non deterministic models and minimal automata).

Apply Knowledge and understanding: at the end of the course, students will be able to solve simple problems of automata theory, by using algebraic and combinatorial techniques: construction of automata for the acceptance of languages, decidability and algorithmic properties of automata, tools to verify the non-recognazibility of formal languages.

Analytical and judgment abilities: successful students will be able to manipulate the basic objects of the theory and they will be able to understand the proofs of some theorems that are relevant in the theory of automata. Moreover they will be able to analyse relations with topics of mathematical theory of formal languages and theory of codes.

Communication skills: the student will be able to present, in a written classwork, his knowledge of the theory and the solutions of the exercises.

Learning skills: the acquired knowledge and skills will permit the student to study, at individual level or in a course taught in the LM, more advanced aspects of automata theory and of mathematical theory of formal languages.

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

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

General objectives: to acquire basic knowledge on the theory of simplicial homology and persistent homology.

Specific objectives:

Knowledge and understanding: at the end of the module the student will have acquired the basic notions and results relating to the theory of finitely generated abelian groups, abstract simplicial complexes, homology and possible applications to topological data analysis.

Apply knowledge and understanding: at the end of the module the student will be able to solve simple problems that require the use of homological techniques in the study of group theory and data analysis.

Critical and judgment skills: the student will have the basis to analyze the analogies and relationships between the topics covered and topics of algebraic topology and (acquired in the Algebraic Topology course). The student will also have the basis to approach a part of the literature in topological data analysis in a mathematically correct and formalized manner.

Communication skills: the student will have the ability to correctly expose the course contents to an audience of people with appropriate mathematical knowledge.

Learning skills: the acquired knowledge will allow a study, individual or taught in a PhD course, relating to more advanced aspects of algebraic topology and / or topological data analysis.

Educational objectives

General objectives: to acquire basic knowledge of classical projective geometry and plane algebraic curves.

Specific objectives:

Knowledge and understanding: at the end of the module the student will have acquired the basic notions and results relating to classical projective geometry (projectivity, perspectives, cross-ratio, single-line constructions) and to the theory of plane algebraic curves (Bezout's theorem, singularities, inflections and elliptic curves).

Applying knowledge and understanding: at the end of the module the student will be able to solve simple problems that require the use of geometric techniques in the study of projective spaces and algebraic curves.

Critical and judgment skills: the student will have the basics to analyze the analogies and relationships between the topics covered and topics in the history of mathematics (on the development of projective geometry) and in the use of elliptic curves in cryptography.

Communication skills: the student will have the ability to correctly expose the course contents to an audience of people with appropriate mathematical knowledge.

Learning skills: the acquired knowledge will allow a study, individual or taught in a PhD course, related to more advanced aspects of algebraic geometry and cryptography.

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 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:

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

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 hyperbolic systems and applications in mechanics, like stability theory. Moreover, they will learn part of the general theory of hyperbolic invariant sets, with applications to homoclinic intersections, chaotic motion, and ergodic theory, in the framework of concrete mechanical systems.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study equilibrium stability problems 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:
Successful students will be able to deal with topics concerning Functional Analysis and foundations of the theory of Operator Algebras, will become proficient and acquainted with subjects such as Spectral Theory, closed, hermitian and selfadjoint operators, C* Algebras and von Neumann Algebras.

Skills and attributes:
Successful students will be able to study the structure of linear operators like simple perturbations of the Laplacian and to use the theory of Operator Algebras in many fundamental questions, from Harmonic analysis to Quantum Mechanics.

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 Goals: rigorous knowledge of probabilistic models from
applications to the relationship with other part of mathematic.

Specific goals:

Knowledge and understanding: at the end of the course the student will
have acquired the basic notions and results related to probability
spaces, random variables, independence, laws of large numbers,
characteristic functions, weak convergence, limit theorems.

Apply knowledge and understanding: at the end of the
course the student will be able to solve simple problems that require
the use of probabilistic techniques both in applications and in
problems of pure mathematics.

Critical and judgmental skills: the student will have the basis to
analyze the analogies and the relationships between the topics covered
and topics of analysis or mathematical physics; will also acquire the
tools that have historically led to the solution of classical
problems.

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 study, individual
or given in an LM course, related to more specialized aspects of
probability.

Educational objectives

General targets:
acquire basic specialist knowledge on
some classic topics of Physics-Mathematics.

Knowledge and understanding:
knowledge of the theory of compact self-adjoint operators, of the applications of this theory
to the theory of potential; basic knowledge of Hamiltonian Mechanics
and of Quantum Mechanics.

Applying knowledge and understanding:
the student will be able to
analyze the spectrum of operators, also for unbounbed operators; to determine
the eigenvalues of the Laplacian in domains with symmetries;
translate into Hamiltonian formalism
the Lagrangian problems and solve them for quadatrure;
discuss the solution of the
Schroedinger equation in simple but physically significant cases.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises, subject to written verification.

Making judgements:
ability to enucleate the most significant aspects of the potential theory
and of the theory of motion,
ability to reflect on similarities and differences between the classical case
and the quantum one.

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 face
the mathematical-physcs courses on more specialized subjects,
and will allow the student to understand, even independently,
the physical relevance of mathematical questions discussed in other courses.

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 bases 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

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 specialized knowledge about representation theory of Lie algebras.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired advanced notions and results related to representation theory of finite dimensional Lie algebras and an introduction to homological methods in algebra.

Apply knowledge and understanding: at the end of the course the student will be able to read research articles related to the topics of the course.

Critical and judgment skills: the student will have the basics to analyze the analogies and relationships between the topics covered and their developments in representation theory and
homological algebra.

Communication skills: ability to present topics in seminars using techniques and results addressed in the course.

Learning skills: the knowledge acquired will prepare the students to start a research activity in algebra and geometry.

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 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

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 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

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 Goals: rigorous knowledge of probabilistic models from
applications to the relationship with other part of mathematic.

Specific goals:

Knowledge and understanding: at the end of the course the student will
have acquired the basic notions and results related to probability
spaces, random variables, independence, laws of large numbers,
characteristic functions, weak convergence, limit theorems.

Apply knowledge and understanding: at the end of the
course the student will be able to solve simple problems that require
the use of probabilistic techniques both in applications and in
problems of pure mathematics.

Critical and judgmental skills: the student will have the basis to
analyze the analogies and the relationships between the topics covered
and topics of analysis or mathematical physics; will also acquire the
tools that have historically led to the solution of classical
problems.

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 study, individual
or given in an LM course, related to more specialized aspects of
probability.

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 targets:
acquire basic specialist knowledge on
some classic topics of Physics-Mathematics.

Knowledge and understanding:
knowledge of the theory of compact self-adjoint operators, of the applications of this theory
to the theory of potential; basic knowledge of Hamiltonian Mechanics
and of Quantum Mechanics.

Applying knowledge and understanding:
the student will be able to
analyze the spectrum of operators, also for unbounbed operators; to determine
the eigenvalues of the Laplacian in domains with symmetries;
translate into Hamiltonian formalism
the Lagrangian problems and solve them for quadatrure;
discuss the solution of the
Schroedinger equation in simple but physically significant cases.
To develop these aspects, in the course they are assigned and carried out
appropriate exercises, subject to written verification.

Making judgements:
ability to enucleate the most significant aspects of the potential theory
and of the theory of motion,
ability to reflect on similarities and differences between the classical case
and the quantum one.

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 face
the mathematical-physcs courses on more specialized subjects,
and will allow the student to understand, even independently,
the physical relevance of mathematical questions discussed in other courses.

Educational objectives

General targets: acquisition of basic knowledge in automata theory.

Specific targets:

Knowledge and understanding: at the end of the course, students will be acquainted with the notions of deterministic and complete automaton, recognizable language, non-deterministic automaton, and rational language, together with theorems describing some fundamental properties, of algebraic and combinatorial nature, of such structures (description of languages accepted by finite automata in term of finite index congruences, rational operations in the free semigroup of strings, non deterministic models and minimal automata).

Apply Knowledge and understanding: at the end of the course, students will be able to solve simple problems of automata theory, by using algebraic and combinatorial techniques: construction of automata for the acceptance of languages, decidability and algorithmic properties of automata, tools to verify the non-recognazibility of formal languages.

Analytical and judgment abilities: successful students will be able to manipulate the basic objects of the theory and they will be able to understand the proofs of some theorems that are relevant in the theory of automata. Moreover they will be able to analyse relations with topics of mathematical theory of formal languages and theory of codes.

Communication skills: the student will be able to present, in a written classwork, his knowledge of the theory and the solutions of the exercises.

Learning skills: the acquired knowledge and skills will permit the student to study, at individual level or in a course taught in the LM, more advanced aspects of automata theory and of mathematical theory of formal languages.

Educational objectives

General skills
The main goals of the class are: 1) knowing the biological levels of organization; 2) knowing the main cell models of living organisms; 3) understanding the structure and main functions of eukaryotic cells; 4) understanding the molecular basis of heredity and gene expression.

Specific skills

A) Knowledge and understanding
- Acquiring the basic knowledge on cellular organization of living organisms
- Understanding the structure and main functions of the eukaryotic cell and the differences between animal and plant cells
-Understanding the molecular basis of heredity and gene expression

B) Applying knowledge and understanding
- Understanding that the "cell system" is the basis of all the functions of living organisms
- Understanding the importance of the descriptive approach in biology: knowing how to observe and describe with precision
-Knowing how to apply the knowledge acquired to design simple experiments on cells to be performed with one’s own hands and with minimal laboratory equipment
C) Making judgement
- Knowing how to connect elementary knowledge of mathematics, physics and chemistry with those of biology to understand cellular organization and biological processes at the cellular level
-Have learned the scientific method to acquire new knowledge with a critical sense
- Being a more aware citizen (citizenship skills) on the major issues of our time

D) Communication skills
-Being able to communicate the acquired knowledge to other students and reflect together about group discussion

E) Learning skills
- Knowing how to use knowledge on cellular functioning to explore topics related to health education and bioethics

Educational objectives

General skills
The main goals of the class are: 1) knowing the biological diversity and the principles of classification of organisms, with special reference to the animals; 2) knowing the functions of the body systems; 3) understanding the principle of biological evolution; 4) understanding the basics of ecology.

Specific skills

A) Knowledge and understanding
- Acquiring a basic background on Biodiversity
- Understanding the functioning of the main systems of living organisms.
- Understanding the basic mechanisms of biological evolution and ecology.

B) Applying knowledge and understanding
Acquiring the basic knowledge on shared characters of living organisms (cells, tissues, systems, reproductive functions, evolutionary mechanisms, interactions with the environment)
- Understanding the adaptive features of orgnisms
- Knowing how to apply the knowledge acquired to design simple observational experiments on organisms (e.g., dissections of samples; micrsocopic observations)

C) Making judgement
- Knowing how to connect elementary knowledge (from module I) with those of module II to understand the functioning and the biological processes at the level of organisms and ecosystems
-Have learned the scientific method to acquire new knowledge with a critical sense
- Being a more aware citizen (citizenship skills) on the major issues of our time (e.g. conservation of biodiversity)

D) Communication skills
-Being able to communicate the acquired knowledge to other students and reflect together about group discussion

E) Learning skills
- Knowing how to use knowledge on functioning of organisms and ecosystems to explore topics related to health education and bioethics

Educational objectives

General skills
The main goals of the class are: 1) knowing the biological levels of organization; 2) knowing the main cell models of living organisms; 3) understanding the structure and main functions of eukaryotic cells; 4) understanding the molecular basis of heredity and gene expression.

Specific skills

A) Knowledge and understanding
- Acquiring the basic knowledge on cellular organization of living organisms
- Understanding the structure and main functions of the eukaryotic cell and the differences between animal and plant cells
-Understanding the molecular basis of heredity and gene expression

B) Applying knowledge and understanding
- Understanding that the "cell system" is the basis of all the functions of living organisms
- Understanding the importance of the descriptive approach in biology: knowing how to observe and describe with precision
-Knowing how to apply the knowledge acquired to design simple experiments on cells to be performed with one’s own hands and with minimal laboratory equipment
C) Making judgement
- Knowing how to connect elementary knowledge of mathematics, physics and chemistry with those of biology to understand cellular organization and biological processes at the cellular level
-Have learned the scientific method to acquire new knowledge with a critical sense
- Being a more aware citizen (citizenship skills) on the major issues of our time

D) Communication skills
-Being able to communicate the acquired knowledge to other students and reflect together about group discussion

E) Learning skills
- Knowing how to use knowledge on cellular functioning to explore topics related to health education and bioethics

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

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 hyperbolic systems and applications in mechanics, like stability theory. Moreover, they will learn part of the general theory of hyperbolic invariant sets, with applications to homoclinic intersections, chaotic motion, and ergodic theory, in the framework of concrete mechanical systems.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study equilibrium stability problems 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

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 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:
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 bases 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

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

Know and understand some issues and problems that fall within the field of study of mathematics, physical science and nature.
Learn how to look, recognize and enhance links with the sciences and their history, artefacts and places in cities.
To experiment through the example lived in first person, diversified and active methods of teaching-learning, in which body and movement in the environment are also instruments of knowledge. Enhance the concrete operation linked to thinking and designing active teaching materials. Apply teaching-pedagogical knowledge in the implementation of educational projects. Empower students with respect to the co-construction of their knowledge.
 
Learning outcomes - Acquired knowledge: Recognizing the knowledge of scientific knowledge of artefacts and places in the city. Having had the experience of reading scientific texts (direct texts of scientists, or of history of science or of epistemological character), asking questions also in relation to the involvement of science in the history of an era, in the culture and history of societies, in gender and intercultural problems. To know and understand the methodological and didactic aspects of the proposed experiences and the activities carried out in the course, in relation to the scientific topics addressed.
 
Upon completion of the course the student will have an advanced knowledge of research aspects in the fields of the sciences, such as the steps from the description to the following schematization, to the quantification and research of the causes of an observed phenomenon. He will also have developed historical and epistemological knowledge in the field of science. [Descriptor of Dublin n. 1].
The skills acquired will concern a greater ability to work in a group, to formulate questions with clear language, to reflect on their own learning and their difficulties and uncertainties, to analyze the educational aspects from the point of view of different disciplines involved in educational actions and training. It will have integrated modes of use of one's own body and of one's own sensory capacities among the tools of knowing. [Descriptor of Dublin n. 2].
The transversal competences acquired concern critical and judgmental skills, enhanced by participation in reflection and laboratory activities and the ability to ask questions and use an indicative method. [Descriptor of Dublin n. 3].
The intermediate activities of the course and the final ones in the form of the "Science Stands" organized by the students autonomously, also in groups, and presented to specialists and non-special beneficiaries, will allow to use display skills, choice of questions, materials and problems relevant, also on the basis of the age of the recipients, and to put in place a posteriori evaluation capacity of the proposed actions, from a multidisciplinary point of view. [Descriptor of Dublin n. 4].
To have acquired metareflection capacity on one's own and others' way of dealing with new content and issues related to scientific disciplines, and to face uncertainties and difficulties in understanding so that the student is more able to continue the study independently in the course of the life and deepen the scientific and specific themes of design in education and critically address, with the perspective of complexity, materials related to scientific disciplines. [Descriptor of Dublin n. 5].
 
Learning outcomes - Acquired skills: students who have passed the exam will be able to conceive, plan and evaluate educational interventions and projects through museum visits and in significant anthropic and nature spaces, and to select and discard relevant information to the topics studied in formal and informal contexts

Educational objectives

Know and understand some issues and problems that fall within the field of study of mathematics, physical science and nature.
Learn how to look, recognize and enhance links with the sciences and their history, artefacts and places in cities.
To experiment through the example lived in first person, diversified and active methods of teaching-learning, in which body and movement in the environment are also instruments of knowledge. Enhance the concrete operation linked to thinking and designing active teaching materials. Apply teaching-pedagogical knowledge in the implementation of educational projects. Empower students with respect to the co-construction of their knowledge.
 
Learning outcomes - Acquired knowledge: Recognizing the knowledge of scientific knowledge of artefacts and places in the city. Having had the experience of reading scientific texts (direct texts of scientists, or of history of science or of epistemological character), asking questions also in relation to the involvement of science in the history of an era, in the culture and history of societies, in gender and intercultural problems. To know and understand the methodological and didactic aspects of the proposed experiences and the activities carried out in the course, in relation to the scientific topics addressed.
 
Upon completion of the course the student will have an advanced knowledge of research aspects in the fields of the sciences, such as the steps from the description to the following schematization, to the quantification and research of the causes of an observed phenomenon. He will also have developed historical and epistemological knowledge in the field of science. [Descriptor of Dublin n. 1].
The skills acquired will concern a greater ability to work in a group, to formulate questions with clear language, to reflect on their own learning and their difficulties and uncertainties, to analyze the educational aspects from the point of view of different disciplines involved in educational actions and training. It will have integrated modes of use of one's own body and of one's own sensory capacities among the tools of knowing. [Descriptor of Dublin n. 2].
The transversal competences acquired concern critical and judgmental skills, enhanced by participation in reflection and laboratory activities and the ability to ask questions and use an indicative method. [Descriptor of Dublin n. 3].
The intermediate activities of the course and the final ones in the form of the "Science Stands" organized by the students autonomously, also in groups, and presented to specialists and non-special beneficiaries, will allow to use display skills, choice of questions, materials and problems relevant, also on the basis of the age of the recipients, and to put in place a posteriori evaluation capacity of the proposed actions, from a multidisciplinary point of view. [Descriptor of Dublin n. 4].
To have acquired metareflection capacity on one's own and others' way of dealing with new content and issues related to scientific disciplines, and to face uncertainties and difficulties in understanding so that the student is more able to continue the study independently in the course of the life and deepen the scientific and specific themes of design in education and critically address, with the perspective of complexity, materials related to scientific disciplines. [Descriptor of Dublin n. 5].
 
Learning outcomes - Acquired skills: students who have passed the exam will be able to conceive, plan and evaluate educational interventions and projects through museum visits and in significant anthropic and nature spaces, and to select and discard relevant information to the topics studied in formal and informal contexts