Mathematics

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.

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.

Educational objectives

General Objectives: The course aims to provide students with foundational knowledge on core topics (naive set theory, natural numbers) and introduce primary algebraic structures (groups, rings, modules, fields). Subsequently, the course will develop basic elements of group theory and group actions on sets.
Knowledge and Understanding: Students will gain a mathematical understanding of the proposed topics and a thorough knowledge of the theory covered. Particular emphasis is placed on the comprehension of arguments and rigor in the presentation of concepts and reasoning.
Applying Knowledge and Understanding: Students should be able to produce simple, rigorous proofs of mathematical results and solve unfamiliar problems that are clearly related to the theoretical content of the course.
Making Judgments: The presentation of content and arguments is designed to enhance the student's ability to recognize rigorous proofs and identify fallacious reasoning.
Communication Skills: Students will acquire a strong ability to communicate problems, ideas, and solutions regarding the subject matter in both written and oral forms.
Learning Skills: Students will be able to consciously read an undergraduate-level specialized algebra monograph regarding the topics mentioned above. Upon completion of the course, they will be prepared to attend Algebra I without any preliminary knowledge gaps.

Educational objectives

General objectives: The course aims to provide students with the fundamental programming skills needed to tackle and solve mathematical problems using computational tools. Through lectures and laboratory exercises, students will be guided in learning a basic programming language, such as C, along with the main algorithmic techniques. Many of the proposed examples will concern mathematical problems studied in parallel in other courses, thereby creating a direct link between theory and implementation. Students will develop the ability to understand how theoretical problems can be addressed and solved through numerical procedures and translated into correct and efficient code. The laboratory sessions will also foster an operational understanding of the topics covered in the core mathematics courses, allowing students to explore them from a computational perspective. By the end of the course, students will have acquired autonomy in the analysis, design, and implementation of code, developing a systematic working method oriented toward the verification of results. Through the study of a basic programming language, they will also be well prepared to transition smoothly to a higher-level programming language such as Python or MATLAB.

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.

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.

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.

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.

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.

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.

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.

Educational objectives

General Objectives: The course aims to provide students with fundamental knowledge of ring theory, field extensions, and an introduction to Galois theory.
Knowledge and Understanding: Students will achieve a mathematical understanding of the proposed topics and a comprehensive knowledge of the theoretical framework presented.
Applying Knowledge and Understanding: Students should be able to produce simple, rigorous proofs of mathematical results and solve unfamiliar problems that are clearly related to the theoretical content covered in class.
Making Judgments: The presentation of content and arguments is designed to enhance the student's ability to recognize rigorous proofs and identify fallacious reasoning.
Communication Skills: Students will acquire a strong ability to communicate problems, ideas, and solutions regarding the subject matter in both written and oral forms.
Learning Skills: Students will be able to proficiently read undergraduate-level specialized algebra monographs related to the aforementioned topics. Upon completion of the course, they will be prepared to attend advanced algebra courses as well as geometry-based courses (such as general topology, algebraic topology, and differential geometry) where algebraic language is indispensable.

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.

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.

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.

Educational objectives

General objectives: The course introduces the main numerical methods for solving some fundamental problems in analysis and linear algebra, with particular attention to their implementation in a high-level programming language and to the assessment of computational complexity. An essential aspect of the course will be the constant interplay between the construction of algorithms and the study of their convergence, stability, and computational efficiency. The theoretical lectures are complemented by laboratory sessions, aimed at programming the studied methods, graphically representing the results, and constructing test problems to quantitatively evaluate the characteristics of the methods.

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.

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.

Educational objectives

Educational objectives

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.

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.