1031385 | APPLIED ANALYTICAL MODELS | 1st | 1st | 6 | MAT/05 | ITA |
Educational objectives Educational Goals
General objectives:
Acquire basic knowledge in modeling based on ordinary and partial differential equations, in the contexts presented in the program. In particular, he will be able to treat differential equations for networks of chemical reactions, the spread of epidemics, the kinetics of enzymes, the propagation of nerve impulses; in addition, he will be able to deal with models in which there is also dependence on space with diffusive terms.
Specific objectives:
Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results relating to some classes of ordinary differential equations and partial derivative equations useful for the description of models, mainly in the biochemical and epidemiological fields.
Apply knowledge and understanding: at the end of the course the student will be able to present basic models in the biomathematic field, discussing their properties and characteristics. You will also be able to use the electronic calculator to perform basic numerical simulations of nonlinear differential equations using pre-existing libraries.
Critical and judgmental skills: the student will have the bases to analyze the analogies and relationships between the topics covered and topics acquired in previous courses in the same field, critically recognizing their salient features.
Communication skills: the student will have developed the ability to expose the contents in the oral part of the verification.
Learning skills: the knowledge acquired will allow an individual and collective study of the subsequent LM courses that require modeling skills.
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10595860 | Mathematical methods in Statistical Mechanics | 1st | 1st | 6 | MAT/07, MAT/06 | ITA |
Educational objectives General targets:
acquire basic knowledge on a rigorous approach to statistical equilibrium mechanics.
Applying knowledge and understanding:
knowledge of statistical ensembles, Gibbs measures and thermodynamic functionals; understanding of phase transitions for paradigmatic lattice particle models.
Making judgements:
ability to describe mechanical and thermodynamic behavior of large systems of particles.
Communication skills:
ability to identify the main points of the theory, to be able to illustrate the most interesting elements by using appropriate examples, and to discuss the mathematic details for simple models.
Learning skills:
the acquired knowledge will allow to face advanced studies, i.e. at PhD level, related to equilibrium and non-equilibrium statistical mechanics, and to use the basic tools of statistical mechanics in other contexts.
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10593295 | Calculus of Variations | 1st | 2nd | 6 | MAT/05 | ITA |
Educational objectives General objectives:
Many models in mathematical physics and natural sciences in general have variational principles (the principle of minimal energy, minimal action, ...) which describe their equilibrium configurations and dynamic evolutions.
The aim of the course is to make students aware of the variety of problems that can be addressed with variational techniques and to provide them with the basic tools and mathematical language for analyzing the models arising in natural sciences.
Specific objectives:
Knowledge and understanding:
at the end of the course the student will have acquired the basic notions and results on the direct method of calculus variations, conditions for semicontinuity, asymptotic analysis via Gamma convergence, and she/he will be able to apply this methods in various contexts about which they will be provided the functional bases at least in dimension 1 (integral functionals and Sobolev spaces, geometric functionals and elements of geometric measure theory).
Apply knowledge and understanding:
at the end of the course the student will be able to begin the study of advanced calculus of variations. She/he will also be able to formulate a simple variational model (for example linked to a specific application) and analyze its asymptotic behavior or identify the characteristics that make it a robust model.
Critical and judgmental skills:
The student will have the basics to connect and use tools covered in various moments of his studies ranging from analysis, mathematical physics and to probability. She/he will therefore be able to appreciate the interest of a mathematical question in relation also to its use to answer a question coming from an applied problem.
Communication skills:
ability to rigorously expose the theoretical contents of the course and also ability to formulate the problem under consideration by understanding the role of deriving the right model and its analysis. Ability to explain moreover the results in the language related to the application under consideration, potentially understandable by non expert in calculating variations.
Learning ability:
the acquired knowledge will allow to face a possible master's thesis work in the field of applied mathematics in natural sciences both with a more theoretical approach and in connection with the analysis of a specific model of interest for applications.
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1031365 | DYNAMICAL SYSTEMS | 1st | 2nd | 6 | MAT/07 | ITA |
Educational objectives General targets: To acquire advanced knowledge in the theory of dynamical systems.
Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on hyperbolic systems and applications in mechanics, like stability theory. Moreover, they will learn part of the general theory of hyperbolic invariant sets, with applications to homoclinic intersections, chaotic motion, and ergodic theory, in the framework of concrete mechanical systems.
Applying knowledge and understanding: Students who have passed the exam will be able to: i) study equilibrium stability problems both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.
Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.
Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.
Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.
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10620480 | Operators' algebras | 1st | 2nd | 6 | MAT/05 | ITA |
Educational objectives Knowledge and understanding:
Successful students will be able to deal with topics concerning Functional Analysis and foundations of the theory of Operator Algebras, will become proficient and acquainted with subjects such as Spectral Theory, closed, hermitian and selfadjoint operators, C* Algebras and von Neumann Algebras.
Skills and attributes:
Successful students will be able to study the structure of linear operators like simple perturbations of the Laplacian and to use the theory of Operator Algebras in many fundamental questions, from Harmonic analysis to Quantum Mechanics.
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10593299 | Control Theory | 2nd | 1st | 6 | MAT/05 | ITA |
Educational objectives 1) Knowledge and understanding
At the end of the course, the students will know and understand:
a) the idea of control system and of differential inclusion, and their basic properties;
b) thr idea of optimal control and necessary and/or sufficient conditions for its existence;
c) the relationship between optimal solutions of a control problem and the Hamilton-Jacobi-Bellman equation;
d) the idea of viscosity solution for the Hamilton-Jacobi equation.
2) Applying knowledge and understanding
At the end of the course, the students will be able to:
a) write the mathematical formulation of an optimal control problem;
b) determine, using the Pontryagin Maximum Principle, the optimal solutions of an optimal control problem;
c) analyze, from a theoretical point of view, the solutions of an optimal control problem through the study of the associated Hamilton-Jacobi-Bellman equation.
3) Making judgements
During the lessons, several problems will be proposed to the students.
Thanks to the autonomous resolution of the problems, and the subsequent discussion in the classroom, the students will acquire both the ability to evaluate their knowledge and the ability to tackle a wide range of optimal control problems.
4) Communication skills
The written form of the exercises, assigned either during lessons or during the written test, and the oral exam will allow the students to evaluate their skill in correctly communicating the knowledges acquired during the course.
5) Learning skills
At the end of the course the students will be able to analyze optimal control problems; such skill is acquired by means of several model problems assigned during the course.
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1031358 | SUPERIOR ALGEBRA | 2nd | 1st | 6 | MAT/02 | ITA |
Educational objectives General objectives: to acquire specialized knowledge about representation theory of Lie algebras.
Specific objectives:
Knowledge and understanding: at the end of the course the student will have acquired advanced notions and results related to representation theory of finite dimensional Lie algebras and an introduction to homological methods in algebra.
Apply knowledge and understanding: at the end of the course the student will be able to read research articles related to the topics of the course.
Critical and judgment skills: the student will have the basics to analyze the analogies and relationships between the topics covered and their developments in representation theory and
homological algebra.
Communication skills: ability to present topics in seminars using techniques and results addressed in the course.
Learning skills: the knowledge acquired will prepare the students to start a research activity in algebra and geometry.
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10596056 | Mathematical methods in quantum mechanics | 2nd | 1st | 6 | MAT/07 | ITA |
Educational objectives General skills
The course aims to transmit to students a deep knowledge of the mathematical structure of Quantum Mechanics, of the historical and conceptual path leading to its formulation, and of its relations with other mathematical subjects (as e.g. functional analysis, operator theory, theory of Lie groups and their unitary representations).
Specific skills
A) Knowledge and understanding
After the conclusion of the course, successful students will know and understand the fundamental concepts of Fourier theory, the mathematical analogy between classical mechanics and geometric optics, the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics, and the mathematical structure of Quantum Theory, with a particular emphasis on dynamical aspects (time evolution) and on the analysis of the symmetries of a quantum system (representation of the symmetry group).
B) Applying knowledge and understanding
The general knowledge will be complemented by the application of general concepts to some specific models, and by the ability to analyze symmetries and dynamics of simple quantum systems. Specific simple systems will be analyzed in detail, including the case of a quantum particle in a linear potential, in a harmonic potential, in a uniform magnetic field, and in a Kepler potential (hydrogenoid atom). Successful students will be potentially able to apply the general concepts also to other more complex systems, including non-hydrogenoid atoms, molecules and crystalline solids.
C) Making judgements
The analysis of the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics will make successful students able to autonomously judge the epistemological foundations of a physical theory, and hence to understand its natural range of application and validity. This critical judgement will lead students to privilege an epistemological apophantic approach, with respect to an apodictic one.
Moreover, successful students will be able to autonomously judge the validity of a mathematical statement, through a critical analysis of the hypotheses and of the deductive steps leading to the proof of the statement itself, and to autonomously formulate counterexamples to mathematical statements whenever one of the hypotheses is denied.
D) Communication skills
Successful students will acquire the ability to communicate what has been learned through written themes and oral exams, and to formulate a logically structured speech, with a clear distinction between hypotheses, deduction and thesis.
E) Learning skills
Successful students will acquire the ability to identify the most relevant topics in a subject and to make the logical connections between the topics covered.
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10605830 | Fourier analysis | 2nd | 1st | 6 | MAT/05 | ENG |
Educational objectives General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.
Specific objectives:
Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).
Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.
Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.
Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.
Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.
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10605751 | Stochastic Calculus and Applications | 2nd | 1st | 6 | MAT/06 | ENG |
Educational objectives Knowledge and understanding:Successful students will learn various characterizations of Brownianan motion, the fundamental properties of diffusion processes and the main results of stochastic calculus, including the Ito formula.Skills and attributes:Successful students will be able to apply stochastic calculus in various applications, from mathematical finance to physics and biology.
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10605832 | Advanced Topics in Geometry | 2nd | 1st | 6 | MAT/03 | ENG |
Educational objectives Knowledge and understanding: at the end of the course the student will be acquainted with basic notions and results in the theory of schemes, cohomology of coherent sheaves, and the theory of projective curves and surfaces.
Applying knowledge and understanding: at the end of the course the student will be able to read and comprehend some papers in Algebraic Geometry.
Analytical and judgment abilities: the student will appreciate the analogies between classical Algebraic Geometry and Number Theory.
Communication skills: the student will be able to communicate the contents of the lectures, in particular illustrating them via concrete examples.
Learning skills: the acquired notions will allow the student to study (either by themselves or in a PhD course) more advanced topics in Algebraic Geometry.
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