Mathematics

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Curriculum(s) for 2025 - Mathematics (33592)

Generale

1st year

Lesson	Semester	CFU	SSD	Language
97786 \| LINEAR ALGEBRA	1st	9	MAT/03	ITA
Educational objectives General aim: to acquire basic knowledge on systems of linear equations, vector spaces, linear maps, affine spaces, euclidean spaces. Specific aim: Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems, matrix calculus, vector spaces, linear maps between them, affine and euclidean spaces. Applied knowledge and comprehension: successful students will be able to solve systems of linear equations with a finite number of variables, and to recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them; he will be able to manipulate matrices and determine the solvability of a system of linear equations and the invertibility of a linear map by studying its rank and by computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism and to determine the associated decomposition into eigenspaces; he/she will be able to solve problems in which euclidean products appear; he/she will be able to solve problems involving affine and euclidean spaces; moreover he/she will acquire rudimental knowledge of some basic fundamental algebraic structure, such as groups, which will be investigated in greater depth in later courses. Critical thinking abilities: in this course the student will acquire basic knowledge that will make him able to discover analogies between the topics learnt in the course and topics in group theory (that will be taught in Algebra 1), functions of many real variables (that will be taught in Analisi 2), geometry of quadrics and of projective spaces (that will be taught in Geometria 1). Communication skills: ability to illustrate the contents of the course in the oral exam and, eventually, in answering written theoretical questions. Learning skills: the knowledge acquired will allow the student to approach the study (on an individual basis, or in a LM courses) of the theory of linear operators on vector spaces possibly of infinite dimensione, of family of vector spaces (vector bundles) and of the eigenspace decompositions of commutative algebras of endomorphisms, and of riemannian geometry.
10599697 \| ANALYSIS I	1st	9	MAT/05	ITA
Educational objectives GENERAL OBJECTIVES: to obtain a general knowledge of the basic techniques of Differential and Integral Calculus and of the standard applications to problems of maxima-minima of functions of a real variable, to the study of their graph, to the convergence of numerical series and to the calculus of definite and indefinite integrals. SPECIFIC OBJECTIVES: Knowledge and understanding: at the end of the course, students will master the basic notions of Differential and Integral Calculus, in particular the notions of function, limit, continuity, numerical series, derivatives and definite integrals. Applying knowledge and understanding: students will be able to solve typical problems from Differential and Integral Calculus, such as the explicit calculation of derivatives, of maxima and minima of a function, to plot an approximate graph of functions of a real variable, to determine the convergence of a numerical series and to compute a definite integrals. Critical and judgment skills: students will be able to use a graph as a tool to analyse concrete phenomena which admit a mathematical description. They will also acquire the tools that have historically led to the solution of classic problems and the basic tools needed in other courses of mathematical analysis, numerical analysis and mathematical phisics. Communication skills: ability to display the contents in the oral part of the verification and in any theoretical questions present in the written test. Learning skills: the notions and techinques learned will give the student access to more advanced notions, either in a further course or in the form of self-study, concerning further aspects of Differential and Integral Calculus.
10600132 \| Computational and programming laboratory	1st	9	INF/01, MAT/08	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
Computational laboratory	1st	6	INF/01	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
Programming laboratory	1st	3	MAT/08	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
AAF1101 \| English language	1st	3	N/D, N/D	ITA
Educational objectives General purposes: reaching the level B1 within the parameters of the Common European Framework of Reference for Languages. Specific purposes: Knowledge and comprehension: at the end of the English course students will develop the linguistic skills of the B1 level within the CEFR: • they can understand the main points of a clear and standard speech; • they can understand written texts dealing with everyday or job-related language; • they can cope with different communicative situations like travelling abroad, everyday conversations about topics that are familiar or of personal interest; • they can write simple texts concerning topics which are familiar or of personal interest. Knowledge and comprehension application: at the end of the English course, students will be aware of the grammatical structures and the vocabulary that correspond to the B1 level of the CEFR. Moreover, they will also be able to read and understand written and spoken texts as well. Critical-thinking skills and judgment: students will be able to independently analyze both written and spoken texts within the B1level of the CEFR. Communicative skills: students will be able to independently and simply talk about topics that are familiar or of personal interest and to communicate abroad. Learning skills: students can develop and reinforce the skills acquired during the course in a further English course and reach the level B2 of the CEFR.
10599508 \| ELEMENTS OF REAL ANALYSIS	2nd	9	MAT/05	ITA
Educational objectives General objectives: to acquire basic knowledge of Mathematical Analysis. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired basic knowledge and results about the theory of metric spaces, Banach spaces and some operators acting on them. Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of the principle of contractions, to solve linear differential equations with constant coefficients of the first and second order, study the convergence of sequences and series of functions, continuity and differentiability properties of vector functions. Critical and judgment skills: in the course the student will be in contact with the first elements of the modern Mathematical Analysis, obtaining the knowledge and the skills necessary to recognize the abstract structures that allow you to tackle and solve some mathematical problems (pure or applied). Communication skills: the student will be able to present the theoretical contents learned in the course and to organize and communicate the reasoning necessary to solve theoretical questions proposed during the lectures. Learning skills: the knowledge acquired is necessary to face the successive courses in Mathematical Analysis.
1022430 \| PROBABILITY I	2nd	9	MAT/06	ITA
Educational objectives General objectives: to acquire basic knowledge in probability theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to probability theory on finite and countable spaces, to the concept of random discrete vectors and to the concept of continuous random variable. Applying knowledge and understanding: at the end of the course the student will be able to solve simple problems in discrete probability, problems concerning discrete random vectors and random numbers represented by continuous random variables. The student will also be able to understand the meaning and implications of independence and conditioning (in the context of discrete models), to understand the meaning of some fundamental limit theorems, such as the law of large numbers. Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between the topics of the course with topics of mathematical analysis and combinatorics (acquired in the “Analisi I” course and treated in the course of “Fondamenti di Analisi Reale”). Communication skills: ability to expose the contents of the course in the oral part of the test and in any theoretical questions present in the written test. Learning skills: the acquired knowledge will allow a study, individual or given in a course related to more specialized aspects of probability theory.
1022431 \| GEOMETRY I	2nd	9	MAT/03	ITA
Educational objectives General objectives: acquiring the techniques of diagonalization of quadratic forms and basic knowledge of affine, euclidean and projective geometry. Specific objectives: Knowledge and understanding: at the end of the course students will have acquired basic results on diagonalizability of quadratic forms and of symmetric operators, as well as elemetary notions of affine, euclidean and projective geometry, and of the natural transformations in each of these ambients. Applying knowledge and understanding: at the end of the course students will be able to solve simple problems requiring the use of diagonalizability of quadratic forms, and to solve elementary problems in affine, euclidean and projective geometry. Critical and judgmental skills: students will have acquired the necessary maturity to recongnize the close relationship between linear algebra and geometry, with specific reference to the notions acquired during the Linear Algebra course; they will have also acquired the tools to formulate and solve classical geometry problems in a modern language. Communication skills: ability of exposition with clarity of notions, definitions, theorems and problem solutions during the written and oral part of the exam. Learning skills: the acquired knowledge will allow the students to undertake with maestry the subsequent study of more technical and abstract geometry theories such as topology and differential geometry.

2nd year

Lesson	Semester	CFU	SSD	Language
10600126 \| ALGEBRA I	1st	12	MAT/02, MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
rings and fields	1st	6	MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of elementary number theory and group theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Modular arithmetic. 2) Group theory. Applying knowledge and understanding: at the end of the module the student will be able to autonomously handle the initial techniques of group theory and to solve simple problems of modular arithmetic Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and transformation groups. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
10599698 \| ANALYSIS II	1st	9	MAT/05	ITA
Educational objectives Overall objectives: to acquire the main tools of Mathematical Analysis concerning the functions of several real variables. Specific objectives: Knowledge and understanding: students passing the exam at the end of the course will have a deep knowledge of the main concepts of mathematical analysis related to functions of several variables, with particular attention to differential calculus, to integration theory, to integrability of differential forms, to main theorems, such as divergence and Stokes ones. Apply knowledge and understanding: Students passing the exam at the end of the course will be able to apply various techniques of differential and integral calculus for functions of several variables. In particular, they will be able to compute integrals of functions of two and three variables, and to find extremals, to integrate differential forms or to compute a surface area. Critical and judgmental skills: the student will be able to analyze analogies and relationships between the new topics and those related to the theory of the functions of one real variable (acquired in Calculus I) and to the general theory of metric spaces (acquired in the course of Analysis I); it will also have a first approach to the measure theory that will be explored in the subsequent course of Real Analysis.
1032750 \| GENERAL INFORMATICS	1st	9	INF/01	ITA
Educational objectives General objectives Acquire basic knowledge on the design of basic algorithms, iterative and recursive algorithms, and the computation of their computational efficiency. Specific objectives Knowledge and understanding: At the end of the course students will know the basic methodologies for the design and analysis of iterative and recursive algorithms, the main data structures, some ways to explore such structures, the main sorting algorithms and the most basic implementations of the dictionaries. They will have a good knowledge of the C language, including advanced aspects such as dynamic memory allocation, pointer arithmetic and separate program compilation. Apply knowledge and understanding: At the end of the course students will have become familiar with the main basic data structures, in particular those implementing dictionaries. They will be able to explain the algorithms and analyze their time complexity, highlighting how their performances depend on the used data structure. They will be able to design new data structures and related algorithms, on the basis of the existing ones; they will be able to explain the main sorting algorithms, illustrating the underlying design strategies and their time complexity analysis; they will be able to compare the asymptotic behavior of the execution times of the studied algorithms, to design recursive solutions to problems and to analyze their asymptotic time complexity. Finally, they will be able to implement the learned algorithms and data structures in the C language, with attention also to the correctness, clarity and concrete efficiency of the programs. Critical and judgmental skills: Students will be able to analyze the quality of an algorithm and related data structures, both from the effective resolution of the problem and from the time complexity point of views. Communication skills: Students will acquire the ability to expose their knowledge in a clear and organized way, which will be verified both through the written tests and during the oral examination. Students will be able to express an algorithmic idea rigorously both at high level, through the use of the pseudocode, and in the C language. Learning ability: Once the cycle of studies is completed, the acquired knowledge will allow students to face the study of algorithmic techniques, of more advanced data structures and of advanced programming methodologies within a master's degree course.
1017999 \| GENERAL PHYSICS I	1st	9	FIS/01	ITA
Educational objectives Introduction to the fundamental concepts of Mechanics and Thermodynamics. Successful students will be able to deal with basic topics concerning Mechanics and Thermodynamics. They will become proficient and acquainted with subject such as work, energy and conservation laws. Moreover, they will be able to afford and solve problems of Mechanics and Thermodynamics by applying the main laws of Physics. In order to achieve these goals, and to help the student to develop the capability to communicate the acquired knowledges, and to continue the studies autonomously, we plan to involve him/her, during the theoretical lectures and excercises, through general and specific questions related to the subject, or through the presentation of some specific subject agreed with the teacher.
AAF1299 \| MATLAB	1st	3	N/D	ITA
Educational objectives General goals: to acquire computer programming skills in MATLAB, which is one of the most used languages in numerical calculation, and to apply the acquired computer skills to the resolution of some mathematical problems and for the graphics of data and functions. Specific goals: Knowledge and comprehension: students who have passed the exam will be able to implement simple algorithms and create graphs using MATLAB software. Apply knowledge and comprehension: students will be able to develop codes in the MATLAB environment to solve numerical problems, using the main functions of MATLAB Critical skills and judgment: students will have the basis for creating elementary mathematical algorithms and structuring through vectorization, which proves to be optimal in the MATLAB environment from the point of view of computational efficiency. Learning skills: the knowledge acquired will allow students to study of problems that require scientific programming skills and will certainly facilitate them in learning other software of interest for scientific calculation and for future work.
1022367 \| REAL ANALYSIS	2nd	9	MAT/05	ITA
Educational objectives General targets: to acquire basic knowledge in measure and integration theory, spaces L^p, Fourier series. Specific targets: to acquire the ability to use the definitions and theorems contained in the course. Knowledge and understanding: the student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series. Apply knowledge and understanding: the student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2. Critical and judgmental skills: the student will have the basis to deal with some problems of pure and applied mathematics, related to the Theory of Measurement, the spaces L^p, the spaces of Hilbert, the Fourier series. Communication skills: the student will be able to expose the course contents in a clear and understandable way, both in the written and oral part. Learning skills: the knowledge acquired will allow study, individual or taught in a master-level course, relating to more specialized aspects concerning the Theory of Measurement, the spaces L ^ p, the spaces of Hilbert, the Fourier series.
10600126 \| ALGEBRA I	2nd	12	MAT/02, MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
arithmetic and Groups	2nd	6	MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
1023149 \| GEOMETRY II	2nd	9	MAT/03	ITA
Educational objectives General objectives: to acquire basic knowledge in general topology, with an introduction to algebraic topology and differential geometry. Specific objectives: Knowledge and understanding: At the end of the course the student will have acquired the concepts and the results basic general topology, with various possible approaches to the notions of topological space, continuous application, homeomorphism; then constructions of topologies on subspaces, products and quotients, topological properties of separation, numerability, compactness, and connection connection for arches. The student will also have acquired the notion of fundamental group and the its use together with the relevant calculation techniques, and the fundamental elements of the theory of topological coatings. Finally, the student will have acquired the basics of geometry differential of curves and surfaces in three-dimensional Euclidean space. Apply knowledge and understanding: At the end of the course the student will be able to solve simple topology problems, even with the use of elementary algebraic topology. He will also know use the notion of curvature in the contexts of the differential geometry of the curves and of the surfaces. Critical and judgmental skills: The student will have the basis for analyzing the similarities and relations between the topics covered and the fundamental notions of the theory of continuity and differentiability, also with tools that have historically led to the solution of classical problems. 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 ability: The acquired knowledge will allow a study, individual or given in a subsequent three-year or master's degree course, related to more advanced aspects of geometry.
1001746 \| RATIONAL MECHANICS	2nd	9	MAT/07	ITA
Educational objectives General targets: To acquire basic knowledge in classical mechanics. Knowledge and understanding: Students who have passed the exam will be able to construct mathematical models not only for problems of mechanical nature, and to use analytic and qualitative methods of ordinary differential equations to deal with them. Applying knowledge and understanding: Students who have passed the exam will be able: i) to perform the qualitative analysis on the phase space for one-dimensional conservative systems and to obtain quantitative estimates; ii) to study problems of stability of equilibrium points elementary methods of Liapunov; iii) to calculate frequencies of normal modes around stable equilibria; iv) to choose properly Lagrangian coordinates for particular configuration manifolds (like Euler angles for SO(3), spherical coordinates, etc.); iv) to recognize the variational nature of Lagrange equations and their implications; v) to use specific criteria for searching prime integrals in Lagrange equations and to perform the subsequent reduction to a smaller number of degrees of freedom. Making judgements: Students who have passed the exam will have the basis to analyze the similarities between the topics covered in the course and the already acquired knowledges in analysis and geometry; they will also acquire important tools and ideas that have historically led to the solution of fundamental problems of classical mechanics. Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of analytical mechanics. Learning skills: The acquired knowledge will allow students who have passed the exam to face the study, at an individual level or in a master's degree course, of specialized aspects of classical mechanics and, more generally, of the theory of dynamical systems.

3rd year

Lesson	Semester	CFU	SSD	Language
1035142 \| GENERAL PHYSICS II	1st	9	FIS/02	ITA
Educational objectives The aim of the course is to provide the basic theoretical understanding of classical electromagnetism. Expected results: the ability to lay out and solve standard exercises on electrostatic, magnetostatic, slowly varying electric and magnetic fields. Acquired knowledge: after passing the exam, the students will be able to profitably follow advanced courses in theoretical physics. Acquired competences: besides learning the fundamental physics laws of electromagnetism, the students will develop the specific skills needed to address and solve scientific problems via analytical methods, in order to study, model and understand classical electromagnetic phenomena.
1022388 \| MATHEMATICAL PHYSICS	1st	9	MAT/07	ITA
Educational objectives General objectives: To acquire basic knowledge on modeling and solving classical problems of continuum physics. Specific objectives: Knowledge and understanding: At the end of the course the student will know the fundamental equations of mathematical physics (transport, waves, Laplace, heat), their derivation from concrete physical problems and the classical techniques of solution. Applying knowledge and understanding: Students who have passed the exam will be able to solve transport and Liouville's equation, simple initial and boundary value problems for wave and heat equations and boundary value problems for Laplace and Poisson equations, using the classical techniques of mathematical physics, like Green's functions and Fourier method. Making judgments : Students who have passed the exam will be able to recognize a mathematical physics approach to problems, linking the mathematical properties of the models based on partial differential equations to the concrete description of the problems of continuum physics. Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of Mathematical Physics related to continuum physics. Learning skills: The acquired knowledge will allow a study, individual or given in other courses, concerning more advanced methods of Mathematical Physics.
Elective course	1st	6	N/D	ITA
Elective course	2nd	6	N/D	ITA
AAF1007 \| Final exam	2nd	9	N/D	ITA
Educational objectives The final exam for the attainment of the Degree consists in the preparation and discussion, in front of a special commission, of an individual written paper, prepared by the student under the supervision of at least one teacher.
AAF2503 \| Mathematical Statistics Lab	2nd	3	N/D	ITA
Educational objectives The main educational objective is an introduction to the fundamental tools of statistics, both in theoretical and computational aspects. Through the use of appropriate software, these tools will be applied in the analysis of real and simulated data sets.
THREE-DIMENSIONAL MODELING

Optional groups

THREE-DIMENSIONAL MODELING: The student must acquire 12 CFU from the following exams

Lesson	Year	Semester	CFU	SSD	Language
1021796 \| Differential geometry	3rd	1st	6	MAT/03	ITA
Educational objectives LEARNING RESULT - ACQUIRED KNOWLEDGE: -know the basic concepts of the differential geometry of the differentiable manifolds of R ^ n and of the differential calculus on them. -know the basic results on curves and surfaces. -know the notion of abstract Riemannian surfaces and manifolds, and the two fundamental theorems of Gauss and Gauss-Bonnet. -know the basic elements of spherical and hyperbolic geometry.
1051938 \| Algebra II	3rd	1st	6	MAT/02	ITA
Educational objectives General objectives: to develop the knowledge of Group Theory and Field Theory and acquire the knowledge of Galois Theory, with the natural applications to the solution of algebraic equations. Specific objectives: at the end of the course the student will have acquired basic knowledge and results concerning Galois Theory. Apply knowledge and understanding: the student will be able to solve problems of algebraic-combinational type requiring the use of techniques related to the presented topics. Critical and judgment skills: at the end of the course the student will have the basis for conducting him towards more complex questions in the framework of modern Algebra. Communication skills: the student will have the ability to communicate rigorously ideas and contents shown in the course. Learning skills: the acquired knowledge will allow to carry on an autonomous study in a possible interdisciplinary context.
1022383 \| DIFFERENTIAL EQUATIONS	3rd	1st	6	MAT/05	ITA
Educational objectives Knowledge and understanding: Concerning equations and systems of ordinary differential equations (ODE) students will learn under which hypotheses solutions exist and when they can be given explicitly, how to investigate the properties of solutions and their maximal existence interval when they are not explicit. The knowledge of proofs gives students a deep understanding of the subject and of the role of hypotheses. Skills and attributes: Techniques for explicit determination of the solutions of the most common ODEs and of linear systems of differential equations with constant coefficients. Qualitative analysis of some systems of ODEs. Applying knowledge and understanding: Students will be able to study the behavior of solutions of some simple ODEs in different application fields. Communication skills: In view of the oral exam students will be able to present a rigorous mathematical proof. Learning skills: Having a complete understanding of the basic mathematical tools used in ODEs, will help students to understand their generalizations for solving more advanced problems. The proof of the existence theorem will help in studying numerical methods for solving ODEs.
1022448 \| ALGEBRA III	3rd	2nd	6	MAT/02	ITA
Educational objectives General objectives: acquire knowledge in commutative algebra and algebraic number theory Specific objectives Knowledge and understanding At the end of the course students will have acquired the main basic notions of Commutative Algebra, concerning commutative rings with or without zero dividers, whole ring extensions, tensor products e flatness, rings and Artinian and Noetherian modules, including the dimension theory for k-algebra finitely generated, primary decomposition, Dedekind domains, ramifications, class number Ability to apply knowledge and understanding At the end of the course students will be able to apply the knowledge acquired competently and thoughtful and solve simple problems that require the use of techniques related to Commutative Algebra and algebraic theory of numbers. Judgment autonomy The student will have the basis for analyzing the analogies and relationships between the topics covered and topics of Geometry or Algebraic Number Theory and will have an idea of how important these are branches of Mathematics are also historically deeply linked. Communication skills • The student will be able to present the main theorems with their proofs in the context of oral test and will be able to communicate the key ideas of Algebra to non-specialists Commutative and algebraic number theory. . Learning ability The student will be able to put to use the topics of Commutative Algebra and algebraic theory of numbers learned in the numerous mathematical contexts in which they are used, both in the context of the Master's degree courses, and in a future research activity
1022957 \| COMPLEX VARIABLE	3rd	2nd	6	MAT/05, MAT/03	ITA
Educational objectives General objectives: to acquire basic knowledge in the theory of a complex variable. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to holomorphic and meromorphic functions, the problem of the calculation of residues and numerous applications, to the analysis of infinite products and some fundamental functions. 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 the complex variable; will be able to calculate improper integrals and to study some special examples of holomorphic and meromorphic functions. Critical and judgmental skills: the student will have the bases to analyze the similarities and relationships between the topics covered and topics of analysis (acquired in the course of Analysis II) or geometry (which can be acquired in the course of Geometry II); 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 the complex variable and the analytic number theory.
THREE-DIMENSIONAL MODELING	3rd	2nd	3	MAT/05	ITA
Educational objectives General objectives: to acquire basic knowledge in the theory of a complex variable. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to holomorphic and meromorphic functions, the problem of the calculation of residues and numerous applications, to the analysis of infinite products and some fundamental functions. 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 the complex variable; will be able to calculate improper integrals and to study some special examples of holomorphic and meromorphic functions. Critical and judgmental skills: the student will have the bases to analyze the similarities and relationships between the topics covered and topics of analysis (acquired in the course of Analysis II) or geometry (which can be acquired in the course of Geometry II); 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 the complex variable and the analytic number theory.
THREE-DIMENSIONAL MODELING	3rd	2nd	3	MAT/03	ITA
10611767 \| CONVEX ANALYSIS	3rd	2nd	6	MAT/05	ITA
Educational objectives Knowledge and understanding Aim of the course is to provide basic notions and skills of Convex Analysis in finite dimensional spaces. Particular attention will be devoted to the analytical aspects of convexity and their use in geometric and optimization problems. Application skills To be able to deal with problems involving convexity, in particular: characterizations of convexity, convex inequalities, regularity and monotony properties of convex functions, subdifferentials, separation of convex sets, convex optimization. Autonomy of judgment To have essential tools for successive approach to functional analysis, partial differential equations, control theory and mathematical programming. To be able to autonomously solve new problems, applying mathematical tools to phenomena or processes to be encountered in University studies or subsequent working activities. Communication skills To know how to communicate using properly mathematical language. Ability to learn To be able to deepen autonomously some arguments introduced during the course.

Storia, didattica e fondamenti

1st year

Lesson	Semester	CFU	SSD	Language
97786 \| LINEAR ALGEBRA	1st	9	MAT/03	ITA
Educational objectives General aim: to acquire basic knowledge on systems of linear equations, vector spaces, linear maps, affine spaces, euclidean spaces. Specific aim: Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems, matrix calculus, vector spaces, linear maps between them, affine and euclidean spaces. Applied knowledge and comprehension: successful students will be able to solve systems of linear equations with a finite number of variables, and to recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them; he will be able to manipulate matrices and determine the solvability of a system of linear equations and the invertibility of a linear map by studying its rank and by computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism and to determine the associated decomposition into eigenspaces; he/she will be able to solve problems in which euclidean products appear; he/she will be able to solve problems involving affine and euclidean spaces; moreover he/she will acquire rudimental knowledge of some basic fundamental algebraic structure, such as groups, which will be investigated in greater depth in later courses. Critical thinking abilities: in this course the student will acquire basic knowledge that will make him able to discover analogies between the topics learnt in the course and topics in group theory (that will be taught in Algebra 1), functions of many real variables (that will be taught in Analisi 2), geometry of quadrics and of projective spaces (that will be taught in Geometria 1). Communication skills: ability to illustrate the contents of the course in the oral exam and, eventually, in answering written theoretical questions. Learning skills: the knowledge acquired will allow the student to approach the study (on an individual basis, or in a LM courses) of the theory of linear operators on vector spaces possibly of infinite dimensione, of family of vector spaces (vector bundles) and of the eigenspace decompositions of commutative algebras of endomorphisms, and of riemannian geometry.
10599697 \| ANALYSIS I	1st	9	MAT/05	ITA
Educational objectives GENERAL OBJECTIVES: to obtain a general knowledge of the basic techniques of Differential and Integral Calculus and of the standard applications to problems of maxima-minima of functions of a real variable, to the study of their graph, to the convergence of numerical series and to the calculus of definite and indefinite integrals. SPECIFIC OBJECTIVES: Knowledge and understanding: at the end of the course, students will master the basic notions of Differential and Integral Calculus, in particular the notions of function, limit, continuity, numerical series, derivatives and definite integrals. Applying knowledge and understanding: students will be able to solve typical problems from Differential and Integral Calculus, such as the explicit calculation of derivatives, of maxima and minima of a function, to plot an approximate graph of functions of a real variable, to determine the convergence of a numerical series and to compute a definite integrals. Critical and judgment skills: students will be able to use a graph as a tool to analyse concrete phenomena which admit a mathematical description. They will also acquire the tools that have historically led to the solution of classic problems and the basic tools needed in other courses of mathematical analysis, numerical analysis and mathematical phisics. Communication skills: ability to display the contents in the oral part of the verification and in any theoretical questions present in the written test. Learning skills: the notions and techinques learned will give the student access to more advanced notions, either in a further course or in the form of self-study, concerning further aspects of Differential and Integral Calculus.
10600132 \| Computational and programming laboratory	1st	9	INF/01, MAT/08	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
Computational laboratory	1st	6	INF/01	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
Programming laboratory	1st	3	MAT/08	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
AAF1101 \| English language	1st	3	N/D, N/D	ITA
Educational objectives General purposes: reaching the level B1 within the parameters of the Common European Framework of Reference for Languages. Specific purposes: Knowledge and comprehension: at the end of the English course students will develop the linguistic skills of the B1 level within the CEFR: • they can understand the main points of a clear and standard speech; • they can understand written texts dealing with everyday or job-related language; • they can cope with different communicative situations like travelling abroad, everyday conversations about topics that are familiar or of personal interest; • they can write simple texts concerning topics which are familiar or of personal interest. Knowledge and comprehension application: at the end of the English course, students will be aware of the grammatical structures and the vocabulary that correspond to the B1 level of the CEFR. Moreover, they will also be able to read and understand written and spoken texts as well. Critical-thinking skills and judgment: students will be able to independently analyze both written and spoken texts within the B1level of the CEFR. Communicative skills: students will be able to independently and simply talk about topics that are familiar or of personal interest and to communicate abroad. Learning skills: students can develop and reinforce the skills acquired during the course in a further English course and reach the level B2 of the CEFR.
10599508 \| ELEMENTS OF REAL ANALYSIS	2nd	9	MAT/05	ITA
Educational objectives General objectives: to acquire basic knowledge of Mathematical Analysis. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired basic knowledge and results about the theory of metric spaces, Banach spaces and some operators acting on them. Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of the principle of contractions, to solve linear differential equations with constant coefficients of the first and second order, study the convergence of sequences and series of functions, continuity and differentiability properties of vector functions. Critical and judgment skills: in the course the student will be in contact with the first elements of the modern Mathematical Analysis, obtaining the knowledge and the skills necessary to recognize the abstract structures that allow you to tackle and solve some mathematical problems (pure or applied). Communication skills: the student will be able to present the theoretical contents learned in the course and to organize and communicate the reasoning necessary to solve theoretical questions proposed during the lectures. Learning skills: the knowledge acquired is necessary to face the successive courses in Mathematical Analysis.
1022431 \| GEOMETRY I	2nd	9	MAT/03	ITA
Educational objectives General objectives: acquiring the techniques of diagonalization of quadratic forms and basic knowledge of affine, euclidean and projective geometry. Specific objectives: Knowledge and understanding: at the end of the course students will have acquired basic results on diagonalizability of quadratic forms and of symmetric operators, as well as elemetary notions of affine, euclidean and projective geometry, and of the natural transformations in each of these ambients. Applying knowledge and understanding: at the end of the course students will be able to solve simple problems requiring the use of diagonalizability of quadratic forms, and to solve elementary problems in affine, euclidean and projective geometry. Critical and judgmental skills: students will have acquired the necessary maturity to recongnize the close relationship between linear algebra and geometry, with specific reference to the notions acquired during the Linear Algebra course; they will have also acquired the tools to formulate and solve classical geometry problems in a modern language. Communication skills: ability of exposition with clarity of notions, definitions, theorems and problem solutions during the written and oral part of the exam. Learning skills: the acquired knowledge will allow the students to undertake with maestry the subsequent study of more technical and abstract geometry theories such as topology and differential geometry.
1022430 \| PROBABILITY I	2nd	9	MAT/06	ITA
Educational objectives General objectives: to acquire basic knowledge in probability theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to probability theory on finite and countable spaces, to the concept of random discrete vectors and to the concept of continuous random variable. Applying knowledge and understanding: at the end of the course the student will be able to solve simple problems in discrete probability, problems concerning discrete random vectors and random numbers represented by continuous random variables. The student will also be able to understand the meaning and implications of independence and conditioning (in the context of discrete models), to understand the meaning of some fundamental limit theorems, such as the law of large numbers. Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between the topics of the course with topics of mathematical analysis and combinatorics (acquired in the “Analisi I” course and treated in the course of “Fondamenti di Analisi Reale”). Communication skills: ability to expose the contents of the course in the oral part of the test and in any theoretical questions present in the written test. Learning skills: the acquired knowledge will allow a study, individual or given in a course related to more specialized aspects of probability theory.

2nd year

Lesson	Semester	CFU	SSD	Language
10600126 \| ALGEBRA I	1st	12	MAT/02, MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
rings and fields	1st	6	MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of elementary number theory and group theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Modular arithmetic. 2) Group theory. Applying knowledge and understanding: at the end of the module the student will be able to autonomously handle the initial techniques of group theory and to solve simple problems of modular arithmetic Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and transformation groups. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
10599698 \| ANALYSIS II	1st	9	MAT/05	ITA
Educational objectives Overall objectives: to acquire the main tools of Mathematical Analysis concerning the functions of several real variables. Specific objectives: Knowledge and understanding: students passing the exam at the end of the course will have a deep knowledge of the main concepts of mathematical analysis related to functions of several variables, with particular attention to differential calculus, to integration theory, to integrability of differential forms, to main theorems, such as divergence and Stokes ones. Apply knowledge and understanding: Students passing the exam at the end of the course will be able to apply various techniques of differential and integral calculus for functions of several variables. In particular, they will be able to compute integrals of functions of two and three variables, and to find extremals, to integrate differential forms or to compute a surface area. Critical and judgmental skills: the student will be able to analyze analogies and relationships between the new topics and those related to the theory of the functions of one real variable (acquired in Calculus I) and to the general theory of metric spaces (acquired in the course of Analysis I); it will also have a first approach to the measure theory that will be explored in the subsequent course of Real Analysis.
1032750 \| GENERAL INFORMATICS	1st	9	INF/01	ITA
Educational objectives General objectives Acquire basic knowledge on the design of basic algorithms, iterative and recursive algorithms, and the computation of their computational efficiency. Specific objectives Knowledge and understanding: At the end of the course students will know the basic methodologies for the design and analysis of iterative and recursive algorithms, the main data structures, some ways to explore such structures, the main sorting algorithms and the most basic implementations of the dictionaries. They will have a good knowledge of the C language, including advanced aspects such as dynamic memory allocation, pointer arithmetic and separate program compilation. Apply knowledge and understanding: At the end of the course students will have become familiar with the main basic data structures, in particular those implementing dictionaries. They will be able to explain the algorithms and analyze their time complexity, highlighting how their performances depend on the used data structure. They will be able to design new data structures and related algorithms, on the basis of the existing ones; they will be able to explain the main sorting algorithms, illustrating the underlying design strategies and their time complexity analysis; they will be able to compare the asymptotic behavior of the execution times of the studied algorithms, to design recursive solutions to problems and to analyze their asymptotic time complexity. Finally, they will be able to implement the learned algorithms and data structures in the C language, with attention also to the correctness, clarity and concrete efficiency of the programs. Critical and judgmental skills: Students will be able to analyze the quality of an algorithm and related data structures, both from the effective resolution of the problem and from the time complexity point of views. Communication skills: Students will acquire the ability to expose their knowledge in a clear and organized way, which will be verified both through the written tests and during the oral examination. Students will be able to express an algorithmic idea rigorously both at high level, through the use of the pseudocode, and in the C language. Learning ability: Once the cycle of studies is completed, the acquired knowledge will allow students to face the study of algorithmic techniques, of more advanced data structures and of advanced programming methodologies within a master's degree course.
1017999 \| GENERAL PHYSICS I	1st	9	FIS/01	ITA
Educational objectives Introduction to the fundamental concepts of Mechanics and Thermodynamics. Successful students will be able to deal with basic topics concerning Mechanics and Thermodynamics. They will become proficient and acquainted with subject such as work, energy and conservation laws. Moreover, they will be able to afford and solve problems of Mechanics and Thermodynamics by applying the main laws of Physics. In order to achieve these goals, and to help the student to develop the capability to communicate the acquired knowledges, and to continue the studies autonomously, we plan to involve him/her, during the theoretical lectures and excercises, through general and specific questions related to the subject, or through the presentation of some specific subject agreed with the teacher.
AAF1299 \| MATLAB	1st	3	N/D	ITA
Educational objectives General goals: to acquire computer programming skills in MATLAB, which is one of the most used languages in numerical calculation, and to apply the acquired computer skills to the resolution of some mathematical problems and for the graphics of data and functions. Specific goals: Knowledge and comprehension: students who have passed the exam will be able to implement simple algorithms and create graphs using MATLAB software. Apply knowledge and comprehension: students will be able to develop codes in the MATLAB environment to solve numerical problems, using the main functions of MATLAB Critical skills and judgment: students will have the basis for creating elementary mathematical algorithms and structuring through vectorization, which proves to be optimal in the MATLAB environment from the point of view of computational efficiency. Learning skills: the knowledge acquired will allow students to study of problems that require scientific programming skills and will certainly facilitate them in learning other software of interest for scientific calculation and for future work.
10600126 \| ALGEBRA I	2nd	12	MAT/02, MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
arithmetic and Groups	2nd	6	MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
1022367 \| REAL ANALYSIS	2nd	9	MAT/05	ITA
Educational objectives General targets: to acquire basic knowledge in measure and integration theory, spaces L^p, Fourier series. Specific targets: to acquire the ability to use the definitions and theorems contained in the course. Knowledge and understanding: the student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series. Apply knowledge and understanding: the student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2. Critical and judgmental skills: the student will have the basis to deal with some problems of pure and applied mathematics, related to the Theory of Measurement, the spaces L^p, the spaces of Hilbert, the Fourier series. Communication skills: the student will be able to expose the course contents in a clear and understandable way, both in the written and oral part. Learning skills: the knowledge acquired will allow study, individual or taught in a master-level course, relating to more specialized aspects concerning the Theory of Measurement, the spaces L ^ p, the spaces of Hilbert, the Fourier series.
1023149 \| GEOMETRY II	2nd	9	MAT/03	ITA
Educational objectives General objectives: to acquire basic knowledge in general topology, with an introduction to algebraic topology and differential geometry. Specific objectives: Knowledge and understanding: At the end of the course the student will have acquired the concepts and the results basic general topology, with various possible approaches to the notions of topological space, continuous application, homeomorphism; then constructions of topologies on subspaces, products and quotients, topological properties of separation, numerability, compactness, and connection connection for arches. The student will also have acquired the notion of fundamental group and the its use together with the relevant calculation techniques, and the fundamental elements of the theory of topological coatings. Finally, the student will have acquired the basics of geometry differential of curves and surfaces in three-dimensional Euclidean space. Apply knowledge and understanding: At the end of the course the student will be able to solve simple topology problems, even with the use of elementary algebraic topology. He will also know use the notion of curvature in the contexts of the differential geometry of the curves and of the surfaces. Critical and judgmental skills: The student will have the basis for analyzing the similarities and relations between the topics covered and the fundamental notions of the theory of continuity and differentiability, also with tools that have historically led to the solution of classical problems. 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 ability: The acquired knowledge will allow a study, individual or given in a subsequent three-year or master's degree course, related to more advanced aspects of geometry.
1001746 \| RATIONAL MECHANICS	2nd	9	MAT/07	ITA
Educational objectives General targets: To acquire basic knowledge in classical mechanics. Knowledge and understanding: Students who have passed the exam will be able to construct mathematical models not only for problems of mechanical nature, and to use analytic and qualitative methods of ordinary differential equations to deal with them. Applying knowledge and understanding: Students who have passed the exam will be able: i) to perform the qualitative analysis on the phase space for one-dimensional conservative systems and to obtain quantitative estimates; ii) to study problems of stability of equilibrium points elementary methods of Liapunov; iii) to calculate frequencies of normal modes around stable equilibria; iv) to choose properly Lagrangian coordinates for particular configuration manifolds (like Euler angles for SO(3), spherical coordinates, etc.); iv) to recognize the variational nature of Lagrange equations and their implications; v) to use specific criteria for searching prime integrals in Lagrange equations and to perform the subsequent reduction to a smaller number of degrees of freedom. Making judgements: Students who have passed the exam will have the basis to analyze the similarities between the topics covered in the course and the already acquired knowledges in analysis and geometry; they will also acquire important tools and ideas that have historically led to the solution of fundamental problems of classical mechanics. Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of analytical mechanics. Learning skills: The acquired knowledge will allow students who have passed the exam to face the study, at an individual level or in a master's degree course, of specialized aspects of classical mechanics and, more generally, of the theory of dynamical systems.

3rd year

Lesson	Semester	CFU	SSD	Language
1035142 \| GENERAL PHYSICS II	1st	9	FIS/02	ITA
Educational objectives The aim of the course is to provide the basic theoretical understanding of classical electromagnetism. Expected results: the ability to lay out and solve standard exercises on electrostatic, magnetostatic, slowly varying electric and magnetic fields. Acquired knowledge: after passing the exam, the students will be able to profitably follow advanced courses in theoretical physics. Acquired competences: besides learning the fundamental physics laws of electromagnetism, the students will develop the specific skills needed to address and solve scientific problems via analytical methods, in order to study, model and understand classical electromagnetic phenomena.
1022365 \| MATHEMATICAL LOGIC	1st	6	MAT/01	ITA
Educational objectives General aims: to acquire basic knowledge and skills in mathematical logic and to be able to apply them in various contexts, including teaching. Specific aims: Knowledge and understanding: The successful student will have acquired basic notions and results in mathematical logic: propositional calculus, predicates calculus, Peano arithmetic and incompleteness results. Applying knowledge and understanding: The successful student will be able to solve exercises and problems of mathematical logic; exercises and problems refer to the topics covered, to other mathematical areas, to teaching and learning mathematics, to natural language. S/he will recognize various kinds of formulas in the simplest cases (tautologies, valid formulas, ...). S/he will be able to recognize and apply inference rules. Critical and judgmental skills: The successful student will be familiar with mathematical rigor and formalism. S/he will have reflected on known mathematical contents and on the translation of concepts in axiomatic theories with appropriate languages. S/he will be able to discuss the role of intuition and rigor in teaching mathematics in different situations. Communication skills: The successful student will be able to present subjects and arguments in the oral test, and to explain what s/he learned. Learning skills: The acquired knowledge will allow to study more specialized subjects. The student will be motivated to extend the acquired knowledge.
1022388 \| MATHEMATICAL PHYSICS	1st	9	MAT/07	ITA
Educational objectives General objectives: To acquire basic knowledge on modeling and solving classical problems of continuum physics. Specific objectives: Knowledge and understanding: At the end of the course the student will know the fundamental equations of mathematical physics (transport, waves, Laplace, heat), their derivation from concrete physical problems and the classical techniques of solution. Applying knowledge and understanding: Students who have passed the exam will be able to solve transport and Liouville's equation, simple initial and boundary value problems for wave and heat equations and boundary value problems for Laplace and Poisson equations, using the classical techniques of mathematical physics, like Green's functions and Fourier method. Making judgments : Students who have passed the exam will be able to recognize a mathematical physics approach to problems, linking the mathematical properties of the models based on partial differential equations to the concrete description of the problems of continuum physics. Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of Mathematical Physics related to continuum physics. Learning skills: The acquired knowledge will allow a study, individual or given in other courses, concerning more advanced methods of Mathematical Physics.
Elective course	1st	6	N/D	ITA
1051921 \| History of mathematics	2nd	6	MAT/04	ITA
Educational objectives General goals acquiring general knowledge of the main stages of the historical developement of mathematics from the time of the Greeks to the beginning of the seveteenth century. Specific goals Knowledge and comprehension: at the end of the course the students who will pass the exam will get the basic knowledges and the methodological tools suitable for further studies in the history of mathematics. Apply Knowledge and comprehension: at the end of the course the students who will pass the exam will be able to understand some of the most significant excerpts taken from the writings of Euclide, Archimede, Cavalieri, Torricelli, Descartes, Fermat, Newton e Leibniz (in italian translation) and they will be able to compare the methods employed by their authors with those of contemporary mathematics. They will also appreciate the educational value of an historical approach to mathematics and to apply it to the elaboration of educational modules for school teaching. Critical skills and judgment: the student will get the basic knowledge necessary for appreciating the historical developement of the elementary concepts and techniques of geometry and analysis and for analysing the analogies and the relationships between the topics treated in this course and those treated in the courses of Calcolo I, Analisi Matematica I and Geometria I. Communication skills: ability to explain the content of the course in the oral part of the exam and to summarize the acquired knowledge in the written part. Learning skills: the acquired knowledge will allow an individual study or a study of a course taught at "Laurea Magistrale", concerning more advanced topics in the history of mathematics.
AAF1007 \| Final exam	2nd	9	N/D	ITA
Educational objectives The final exam for the attainment of the Degree consists in the preparation and discussion, in front of a special commission, of an individual written paper, prepared by the student under the supervision of at least one teacher.
Elective course	2nd	6	N/D	ITA
AAF2503 \| Mathematical Statistics Lab	2nd	3	N/D	ITA
Educational objectives The main educational objective is an introduction to the fundamental tools of statistics, both in theoretical and computational aspects. Through the use of appropriate software, these tools will be applied in the analysis of real and simulated data sets.

Matematica per le applicazioni

1st year

Lesson	Semester	CFU	SSD	Language
97786 \| LINEAR ALGEBRA	1st	9	MAT/03	ITA
Educational objectives General aim: to acquire basic knowledge on systems of linear equations, vector spaces, linear maps, affine spaces, euclidean spaces. Specific aim: Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems, matrix calculus, vector spaces, linear maps between them, affine and euclidean spaces. Applied knowledge and comprehension: successful students will be able to solve systems of linear equations with a finite number of variables, and to recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them; he will be able to manipulate matrices and determine the solvability of a system of linear equations and the invertibility of a linear map by studying its rank and by computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism and to determine the associated decomposition into eigenspaces; he/she will be able to solve problems in which euclidean products appear; he/she will be able to solve problems involving affine and euclidean spaces; moreover he/she will acquire rudimental knowledge of some basic fundamental algebraic structure, such as groups, which will be investigated in greater depth in later courses. Critical thinking abilities: in this course the student will acquire basic knowledge that will make him able to discover analogies between the topics learnt in the course and topics in group theory (that will be taught in Algebra 1), functions of many real variables (that will be taught in Analisi 2), geometry of quadrics and of projective spaces (that will be taught in Geometria 1). Communication skills: ability to illustrate the contents of the course in the oral exam and, eventually, in answering written theoretical questions. Learning skills: the knowledge acquired will allow the student to approach the study (on an individual basis, or in a LM courses) of the theory of linear operators on vector spaces possibly of infinite dimensione, of family of vector spaces (vector bundles) and of the eigenspace decompositions of commutative algebras of endomorphisms, and of riemannian geometry.
10599697 \| ANALYSIS I	1st	9	MAT/05	ITA
Educational objectives GENERAL OBJECTIVES: to obtain a general knowledge of the basic techniques of Differential and Integral Calculus and of the standard applications to problems of maxima-minima of functions of a real variable, to the study of their graph, to the convergence of numerical series and to the calculus of definite and indefinite integrals. SPECIFIC OBJECTIVES: Knowledge and understanding: at the end of the course, students will master the basic notions of Differential and Integral Calculus, in particular the notions of function, limit, continuity, numerical series, derivatives and definite integrals. Applying knowledge and understanding: students will be able to solve typical problems from Differential and Integral Calculus, such as the explicit calculation of derivatives, of maxima and minima of a function, to plot an approximate graph of functions of a real variable, to determine the convergence of a numerical series and to compute a definite integrals. Critical and judgment skills: students will be able to use a graph as a tool to analyse concrete phenomena which admit a mathematical description. They will also acquire the tools that have historically led to the solution of classic problems and the basic tools needed in other courses of mathematical analysis, numerical analysis and mathematical phisics. Communication skills: ability to display the contents in the oral part of the verification and in any theoretical questions present in the written test. Learning skills: the notions and techinques learned will give the student access to more advanced notions, either in a further course or in the form of self-study, concerning further aspects of Differential and Integral Calculus.
10600132 \| Computational and programming laboratory	1st	9	INF/01, MAT/08	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
Computational laboratory	1st	6	INF/01	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
Programming laboratory	1st	3	MAT/08	ITA
Educational objectives General objectives: to acquire computer programming skills in C/C ++, which is one of the most used languages by developers, and to apply the acquired computer skills to solve mathematical problems. Specific objectives: Knowledge and understanding: the students who have passed the exam will have a basic knowledge of computer programming and machine arithmetic and will be able to understand how to structure relatively simple algorithms to effectively solve mathematical problems. Apply knowledge and understanding: at the end of the course the students will be able to solve, through adequate algorithms, relatively simple mathematical problems. They will also be able to design and implement computer programs that interact appropriately with a potential user. Critical and judgmental skills: the students will have the basis to analyze elementary mathematical algorithms from the point of view of computational efficiency, stability and accuracy. They will also be able to understand that theoretical mathematical results must be reformulated to make them useful in the practice of finite arithmetic calculation. Communication skills: ability to expose and motivate the resolution proposed to certain problems chosen in sessions of exercises in the classroom and in the oral exam at the end of the course. Learning skills: the students will have to become familiar and practical with various elements which concern information technology such as the computer programming language, libraries, compilers, the software available on the Internet that offers an integrated development environment under different operating systems, etc. These skills will certainly allow them to learn more easily the use of other software of interest for scientific calculation and the world of work.
AAF1101 \| English language	1st	3	N/D, N/D	ITA
Educational objectives General purposes: reaching the level B1 within the parameters of the Common European Framework of Reference for Languages. Specific purposes: Knowledge and comprehension: at the end of the English course students will develop the linguistic skills of the B1 level within the CEFR: • they can understand the main points of a clear and standard speech; • they can understand written texts dealing with everyday or job-related language; • they can cope with different communicative situations like travelling abroad, everyday conversations about topics that are familiar or of personal interest; • they can write simple texts concerning topics which are familiar or of personal interest. Knowledge and comprehension application: at the end of the English course, students will be aware of the grammatical structures and the vocabulary that correspond to the B1 level of the CEFR. Moreover, they will also be able to read and understand written and spoken texts as well. Critical-thinking skills and judgment: students will be able to independently analyze both written and spoken texts within the B1level of the CEFR. Communicative skills: students will be able to independently and simply talk about topics that are familiar or of personal interest and to communicate abroad. Learning skills: students can develop and reinforce the skills acquired during the course in a further English course and reach the level B2 of the CEFR.
10599508 \| ELEMENTS OF REAL ANALYSIS	2nd	9	MAT/05	ITA
Educational objectives General objectives: to acquire basic knowledge of Mathematical Analysis. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired basic knowledge and results about the theory of metric spaces, Banach spaces and some operators acting on them. Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of the principle of contractions, to solve linear differential equations with constant coefficients of the first and second order, study the convergence of sequences and series of functions, continuity and differentiability properties of vector functions. Critical and judgment skills: in the course the student will be in contact with the first elements of the modern Mathematical Analysis, obtaining the knowledge and the skills necessary to recognize the abstract structures that allow you to tackle and solve some mathematical problems (pure or applied). Communication skills: the student will be able to present the theoretical contents learned in the course and to organize and communicate the reasoning necessary to solve theoretical questions proposed during the lectures. Learning skills: the knowledge acquired is necessary to face the successive courses in Mathematical Analysis.
1022431 \| GEOMETRY I	2nd	9	MAT/03	ITA
Educational objectives General objectives: acquiring the techniques of diagonalization of quadratic forms and basic knowledge of affine, euclidean and projective geometry. Specific objectives: Knowledge and understanding: at the end of the course students will have acquired basic results on diagonalizability of quadratic forms and of symmetric operators, as well as elemetary notions of affine, euclidean and projective geometry, and of the natural transformations in each of these ambients. Applying knowledge and understanding: at the end of the course students will be able to solve simple problems requiring the use of diagonalizability of quadratic forms, and to solve elementary problems in affine, euclidean and projective geometry. Critical and judgmental skills: students will have acquired the necessary maturity to recongnize the close relationship between linear algebra and geometry, with specific reference to the notions acquired during the Linear Algebra course; they will have also acquired the tools to formulate and solve classical geometry problems in a modern language. Communication skills: ability of exposition with clarity of notions, definitions, theorems and problem solutions during the written and oral part of the exam. Learning skills: the acquired knowledge will allow the students to undertake with maestry the subsequent study of more technical and abstract geometry theories such as topology and differential geometry.
1022430 \| PROBABILITY I	2nd	9	MAT/06	ITA
Educational objectives General objectives: to acquire basic knowledge in probability theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to probability theory on finite and countable spaces, to the concept of random discrete vectors and to the concept of continuous random variable. Applying knowledge and understanding: at the end of the course the student will be able to solve simple problems in discrete probability, problems concerning discrete random vectors and random numbers represented by continuous random variables. The student will also be able to understand the meaning and implications of independence and conditioning (in the context of discrete models), to understand the meaning of some fundamental limit theorems, such as the law of large numbers. Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between the topics of the course with topics of mathematical analysis and combinatorics (acquired in the “Analisi I” course and treated in the course of “Fondamenti di Analisi Reale”). Communication skills: ability to expose the contents of the course in the oral part of the test and in any theoretical questions present in the written test. Learning skills: the acquired knowledge will allow a study, individual or given in a course related to more specialized aspects of probability theory.

2nd year

Lesson	Semester	CFU	SSD	Language
10600126 \| ALGEBRA I	1st	12	MAT/02, MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
rings and fields	1st	6	MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of elementary number theory and group theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Modular arithmetic. 2) Group theory. Applying knowledge and understanding: at the end of the module the student will be able to autonomously handle the initial techniques of group theory and to solve simple problems of modular arithmetic Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and transformation groups. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
10599698 \| ANALYSIS II	1st	9	MAT/05	ITA
Educational objectives Overall objectives: to acquire the main tools of Mathematical Analysis concerning the functions of several real variables. Specific objectives: Knowledge and understanding: students passing the exam at the end of the course will have a deep knowledge of the main concepts of mathematical analysis related to functions of several variables, with particular attention to differential calculus, to integration theory, to integrability of differential forms, to main theorems, such as divergence and Stokes ones. Apply knowledge and understanding: Students passing the exam at the end of the course will be able to apply various techniques of differential and integral calculus for functions of several variables. In particular, they will be able to compute integrals of functions of two and three variables, and to find extremals, to integrate differential forms or to compute a surface area. Critical and judgmental skills: the student will be able to analyze analogies and relationships between the new topics and those related to the theory of the functions of one real variable (acquired in Calculus I) and to the general theory of metric spaces (acquired in the course of Analysis I); it will also have a first approach to the measure theory that will be explored in the subsequent course of Real Analysis.
1032750 \| GENERAL INFORMATICS	1st	9	INF/01	ITA
Educational objectives General objectives Acquire basic knowledge on the design of basic algorithms, iterative and recursive algorithms, and the computation of their computational efficiency. Specific objectives Knowledge and understanding: At the end of the course students will know the basic methodologies for the design and analysis of iterative and recursive algorithms, the main data structures, some ways to explore such structures, the main sorting algorithms and the most basic implementations of the dictionaries. They will have a good knowledge of the C language, including advanced aspects such as dynamic memory allocation, pointer arithmetic and separate program compilation. Apply knowledge and understanding: At the end of the course students will have become familiar with the main basic data structures, in particular those implementing dictionaries. They will be able to explain the algorithms and analyze their time complexity, highlighting how their performances depend on the used data structure. They will be able to design new data structures and related algorithms, on the basis of the existing ones; they will be able to explain the main sorting algorithms, illustrating the underlying design strategies and their time complexity analysis; they will be able to compare the asymptotic behavior of the execution times of the studied algorithms, to design recursive solutions to problems and to analyze their asymptotic time complexity. Finally, they will be able to implement the learned algorithms and data structures in the C language, with attention also to the correctness, clarity and concrete efficiency of the programs. Critical and judgmental skills: Students will be able to analyze the quality of an algorithm and related data structures, both from the effective resolution of the problem and from the time complexity point of views. Communication skills: Students will acquire the ability to expose their knowledge in a clear and organized way, which will be verified both through the written tests and during the oral examination. Students will be able to express an algorithmic idea rigorously both at high level, through the use of the pseudocode, and in the C language. Learning ability: Once the cycle of studies is completed, the acquired knowledge will allow students to face the study of algorithmic techniques, of more advanced data structures and of advanced programming methodologies within a master's degree course.
1017999 \| GENERAL PHYSICS I	1st	9	FIS/01	ITA
Educational objectives Introduction to the fundamental concepts of Mechanics and Thermodynamics. Successful students will be able to deal with basic topics concerning Mechanics and Thermodynamics. They will become proficient and acquainted with subject such as work, energy and conservation laws. Moreover, they will be able to afford and solve problems of Mechanics and Thermodynamics by applying the main laws of Physics. In order to achieve these goals, and to help the student to develop the capability to communicate the acquired knowledges, and to continue the studies autonomously, we plan to involve him/her, during the theoretical lectures and excercises, through general and specific questions related to the subject, or through the presentation of some specific subject agreed with the teacher.
AAF1299 \| MATLAB	1st	3	N/D	ITA
Educational objectives General goals: to acquire computer programming skills in MATLAB, which is one of the most used languages in numerical calculation, and to apply the acquired computer skills to the resolution of some mathematical problems and for the graphics of data and functions. Specific goals: Knowledge and comprehension: students who have passed the exam will be able to implement simple algorithms and create graphs using MATLAB software. Apply knowledge and comprehension: students will be able to develop codes in the MATLAB environment to solve numerical problems, using the main functions of MATLAB Critical skills and judgment: students will have the basis for creating elementary mathematical algorithms and structuring through vectorization, which proves to be optimal in the MATLAB environment from the point of view of computational efficiency. Learning skills: the knowledge acquired will allow students to study of problems that require scientific programming skills and will certainly facilitate them in learning other software of interest for scientific calculation and for future work.
10600126 \| ALGEBRA I	2nd	12	MAT/02, MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
arithmetic and Groups	2nd	6	MAT/02	ITA
Educational objectives General objectives: acquire the basic knowledge of Algebra related to topics of ring theory and field theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Theory of Rings. 2) Field theory and their extensions. Applying knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of the theory of rings and fields, and to solve simple ones problems relating to rings of polynomials, Euclidean rings, finite extensions Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the fields of real and complex numbers. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.
1022367 \| REAL ANALYSIS	2nd	9	MAT/05	ITA
Educational objectives General targets: to acquire basic knowledge in measure and integration theory, spaces L^p, Fourier series. Specific targets: to acquire the ability to use the definitions and theorems contained in the course. Knowledge and understanding: the student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series. Apply knowledge and understanding: the student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2. Critical and judgmental skills: the student will have the basis to deal with some problems of pure and applied mathematics, related to the Theory of Measurement, the spaces L^p, the spaces of Hilbert, the Fourier series. Communication skills: the student will be able to expose the course contents in a clear and understandable way, both in the written and oral part. Learning skills: the knowledge acquired will allow study, individual or taught in a master-level course, relating to more specialized aspects concerning the Theory of Measurement, the spaces L ^ p, the spaces of Hilbert, the Fourier series.
1023149 \| GEOMETRY II	2nd	9	MAT/03	ITA
Educational objectives General objectives: to acquire basic knowledge in general topology, with an introduction to algebraic topology and differential geometry. Specific objectives: Knowledge and understanding: At the end of the course the student will have acquired the concepts and the results basic general topology, with various possible approaches to the notions of topological space, continuous application, homeomorphism; then constructions of topologies on subspaces, products and quotients, topological properties of separation, numerability, compactness, and connection connection for arches. The student will also have acquired the notion of fundamental group and the its use together with the relevant calculation techniques, and the fundamental elements of the theory of topological coatings. Finally, the student will have acquired the basics of geometry differential of curves and surfaces in three-dimensional Euclidean space. Apply knowledge and understanding: At the end of the course the student will be able to solve simple topology problems, even with the use of elementary algebraic topology. He will also know use the notion of curvature in the contexts of the differential geometry of the curves and of the surfaces. Critical and judgmental skills: The student will have the basis for analyzing the similarities and relations between the topics covered and the fundamental notions of the theory of continuity and differentiability, also with tools that have historically led to the solution of classical problems. 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 ability: The acquired knowledge will allow a study, individual or given in a subsequent three-year or master's degree course, related to more advanced aspects of geometry.
1001746 \| RATIONAL MECHANICS	2nd	9	MAT/07	ITA
Educational objectives General targets: To acquire basic knowledge in classical mechanics. Knowledge and understanding: Students who have passed the exam will be able to construct mathematical models not only for problems of mechanical nature, and to use analytic and qualitative methods of ordinary differential equations to deal with them. Applying knowledge and understanding: Students who have passed the exam will be able: i) to perform the qualitative analysis on the phase space for one-dimensional conservative systems and to obtain quantitative estimates; ii) to study problems of stability of equilibrium points elementary methods of Liapunov; iii) to calculate frequencies of normal modes around stable equilibria; iv) to choose properly Lagrangian coordinates for particular configuration manifolds (like Euler angles for SO(3), spherical coordinates, etc.); iv) to recognize the variational nature of Lagrange equations and their implications; v) to use specific criteria for searching prime integrals in Lagrange equations and to perform the subsequent reduction to a smaller number of degrees of freedom. Making judgements: Students who have passed the exam will have the basis to analyze the similarities between the topics covered in the course and the already acquired knowledges in analysis and geometry; they will also acquire important tools and ideas that have historically led to the solution of fundamental problems of classical mechanics. Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of analytical mechanics. Learning skills: The acquired knowledge will allow students who have passed the exam to face the study, at an individual level or in a master's degree course, of specialized aspects of classical mechanics and, more generally, of the theory of dynamical systems.

3rd year

Lesson	Semester	CFU	SSD	Language
1035142 \| GENERAL PHYSICS II	1st	9	FIS/02	ITA
Educational objectives The aim of the course is to provide the basic theoretical understanding of classical electromagnetism. Expected results: the ability to lay out and solve standard exercises on electrostatic, magnetostatic, slowly varying electric and magnetic fields. Acquired knowledge: after passing the exam, the students will be able to profitably follow advanced courses in theoretical physics. Acquired competences: besides learning the fundamental physics laws of electromagnetism, the students will develop the specific skills needed to address and solve scientific problems via analytical methods, in order to study, model and understand classical electromagnetic phenomena.
1022388 \| MATHEMATICAL PHYSICS	1st	9	MAT/07	ITA
Educational objectives General objectives: To acquire basic knowledge on modeling and solving classical problems of continuum physics. Specific objectives: Knowledge and understanding: At the end of the course the student will know the fundamental equations of mathematical physics (transport, waves, Laplace, heat), their derivation from concrete physical problems and the classical techniques of solution. Applying knowledge and understanding: Students who have passed the exam will be able to solve transport and Liouville's equation, simple initial and boundary value problems for wave and heat equations and boundary value problems for Laplace and Poisson equations, using the classical techniques of mathematical physics, like Green's functions and Fourier method. Making judgments : Students who have passed the exam will be able to recognize a mathematical physics approach to problems, linking the mathematical properties of the models based on partial differential equations to the concrete description of the problems of continuum physics. Communication skills: Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of Mathematical Physics related to continuum physics. Learning skills: The acquired knowledge will allow a study, individual or given in other courses, concerning more advanced methods of Mathematical Physics.
Elective course	1st	6	N/D	ITA
AAF1007 \| Final exam	2nd	9	N/D	ITA
Educational objectives The final exam for the attainment of the Degree consists in the preparation and discussion, in front of a special commission, of an individual written paper, prepared by the student under the supervision of at least one teacher.
Elective course	2nd	6	N/D	ITA
AAF2503 \| Mathematical Statistics Lab	2nd	3	N/D	ITA
Educational objectives The main educational objective is an introduction to the fundamental tools of statistics, both in theoretical and computational aspects. Through the use of appropriate software, these tools will be applied in the analysis of real and simulated data sets.
THREE-DIMENSIONAL MODELING
THREE-DIMENSIONAL MODELING

Optional groups

THREE-DIMENSIONAL MODELING: The student must acquire 6 CFU from the following exams

Lesson	Year	Semester	CFU	SSD	Language
1010982 \| Numerical analysis	3rd	1st	6	MAT/08	ITA
Educational objectives The course intends to present numerical methods of approximation for the solution of several mathematical problems that occur in many applications and in mathematical modeling. In particular, the following topics will be treated both from a theoretical and an algorithmic point of view: 1. Solution of systems of linear equations 2. Solution of systems of non-linear equations 3. Approximation of functions and experimental data 4. Numerical quadrature 5. Numerical methods for ordinary differential equations The course includes some Lab sessions to develop codes in MATLAB. 1. Knowledge and understanding The students who have passed the exam will know the most important numerical techniques on the topics presented in the course. 2. Applied knowledge and understanding The students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to code the algorithms in MATLAB. 3. Making judgments The students will be able to evaluate the results produced by their programs and to produce tests and simulations. 4. Communication skills The students will be able to present and explain the proposed solution of some problems and exercises both at the blackboard and using a computer. 5. Learning skills The acquired knowledge will construct the basis to study more specialized aspects of numerical analysis, scientific computing and simulations. The students will become familiar with different concepts and techniques related to the topics presented in the course.
1051922 \| Probability II	3rd	1st	6	MAT/06	ITA
Educational objectives General objectives: to acquire knowledge in Probability theory and improve problem solving ability. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions ergodic and information theory, stochastic domination and percolation theory. Apply knowledge and understanding: at the end of the course the student will be able to solve problems on discrete random fields, concerning ergodic and information theory, stochastic domination and percolation theory. Critical and judgmental skills: the student will have the basis to analyze the similarities and relationships between the topics covered in the course and the discrete probability developed in the first probability course. It will acquire familiarity with key concepts in Probability theory, useful also in other fields. Communication skills: the student must show the ability to present the contents of the course in the oral part of the assessment and in the solution of problems in the written test. Learning skills: the acquired knowledge will allow the student to study in depth some aspects of the theories presented in the course and will facilitate the study into very active research fields.
1038308 \| NUMERICAL METHODS OF APPROXIMATION	3rd	2nd	6	MAT/08	ITA
Educational objectives The course will present to present numerical methods of approximation for the solution of several mathematical problems that occur in many applications and in mathematical modeling. The content of the course varies every year among the following topics: Numerical Linear Algebra Numerical Optimization Numerical methods for ordinary differential equations Approximation theory The topics will be treated both from a theoretical and an algorithmic point of view. The course includes some Lab sessions to develop programming codes in C++ or MATLAB. Knowledge and understanding: Students who have passed the exam will know the main numerical techniques on the topics presented in the course. Applied knowledge and understanding: Students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to implement the algorithms in C++ or MATLAB. Critical and judgmental skills: Students will be able to evaluate the results produced by their programs and to produce tests and simulations. Communication skills: Students will be able to expose and motivate the proposed solution of some problems chosen in class either on the blackboard and/or using a computer. Learning skills: The acquired knowledge will allow to build the bases for a study related to more specialized aspects of numerical analysis, scientific computing and simulations. The student will become familiar with different concepts and techniques related to the topics presented in the course.

THREE-DIMENSIONAL MODELING: The student must acquire 6 CFU from the following exams

Lesson	Year	Semester	CFU	SSD	Language
1010982 \| Numerical analysis	3rd	1st	6	MAT/08	ITA
Educational objectives The course intends to present numerical methods of approximation for the solution of several mathematical problems that occur in many applications and in mathematical modeling. In particular, the following topics will be treated both from a theoretical and an algorithmic point of view: 1. Solution of systems of linear equations 2. Solution of systems of non-linear equations 3. Approximation of functions and experimental data 4. Numerical quadrature 5. Numerical methods for ordinary differential equations The course includes some Lab sessions to develop codes in MATLAB. 1. Knowledge and understanding The students who have passed the exam will know the most important numerical techniques on the topics presented in the course. 2. Applied knowledge and understanding The students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to code the algorithms in MATLAB. 3. Making judgments The students will be able to evaluate the results produced by their programs and to produce tests and simulations. 4. Communication skills The students will be able to present and explain the proposed solution of some problems and exercises both at the blackboard and using a computer. 5. Learning skills The acquired knowledge will construct the basis to study more specialized aspects of numerical analysis, scientific computing and simulations. The students will become familiar with different concepts and techniques related to the topics presented in the course.
1038308 \| NUMERICAL METHODS OF APPROXIMATION	3rd	2nd	6	MAT/08	ITA
Educational objectives The course will present to present numerical methods of approximation for the solution of several mathematical problems that occur in many applications and in mathematical modeling. The content of the course varies every year among the following topics: Numerical Linear Algebra Numerical Optimization Numerical methods for ordinary differential equations Approximation theory The topics will be treated both from a theoretical and an algorithmic point of view. The course includes some Lab sessions to develop programming codes in C++ or MATLAB. Knowledge and understanding: Students who have passed the exam will know the main numerical techniques on the topics presented in the course. Applied knowledge and understanding: Students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to implement the algorithms in C++ or MATLAB. Critical and judgmental skills: Students will be able to evaluate the results produced by their programs and to produce tests and simulations. Communication skills: Students will be able to expose and motivate the proposed solution of some problems chosen in class either on the blackboard and/or using a computer. Learning skills: The acquired knowledge will allow to build the bases for a study related to more specialized aspects of numerical analysis, scientific computing and simulations. The student will become familiar with different concepts and techniques related to the topics presented in the course.

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30/07/2025 - Inizio lezioni

Curriculum(s) for 2025 - Mathematics (33592)

1st year

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3rd year

Optional groups

1st year

2nd year

3rd year

1st year

2nd year

3rd year

Optional groups