Parma 2022/23
Lectures take place at “Dipartimento di Scienze Matematiche, Fisiche e Informatiche”, Mathematics Building, Parco Area delle Scienze 53/A, 43124 Parma (location).
Title and Credits: Numerical methods for Boundary Integral Equations, 6 CFU
Teacher: Alessandra Aimi
Syllabus: The course is principally focused on Boundary Element Methods (BEMs).
Lectures involve: Boundary integral formulation of elliptic, parabolic and hyperbolic problems - Integral operators with weakly singular, strongly singular and hyper-singular kernels - Approximation techniques: collocation and Galerkin BEMs - Quadrature formulas for weakly singular integrals, Cauchy principal value integrals and Hadamard finite part integrals -
Convergence results - Numerical schemes for the generation of the linear system coming from Galerkin BEM discretization.
Knowledge of basic notions in Numerical Analysis and in particular in numerical approximation of partial differential equations is required.
References will be provided directly during the course.
Dates 2022/2023: Lectures will take place in Spring 2023 at the University of Parma for an amount of 24 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and Credits: Infinite Dimensional Analysis, 4 CFU
Teacher: Alessandra Lunardi
Syllabus: This is an introductory course about analysis in abstract Wiener spaces, namely separable Banach or Hilbert spaces endowed with non-degenerate Gaussian measures. Sobolev spaces and spaces of continuous functions will be considered. The basic differential operators (gradient and divergence) will be studied, as well as the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup, that are the Gaussian analogues of the Laplacian and the heat semigroup. The most important functional inequalities in this context, such as Poincaré and logarithmic Sobolev inequalities, will be proved. Hermite polynomials and the Wiener chaos will be described.
The reference book is "Gaussian Measures" by V. Bogachev (Mathematical Surveys and Monographs 62, AMS 1998).
According to the interests of the audience, it is possible to consider only the Hilbert space setting, in which case the reference book is "Second Order Partial Differential Equations in Hilbert Spaces" by G. Da Prato and J. Zabczyk (Cambridge Univ. Press 2002).
In any case, lecture notes prepared by the teacher will be available.
Dates 2022/2023: Lectures will take place in Spring 2023 at the University of Parma for an amount of 24 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and credits: Topics in Representation Theory, 6 CFU
Teacher: Andrea Appel
Syllabus: The course will provide an introduction to an advanced topic in representation theory and quantum groups. These include:
1) Categorified quantum groups.
2) Quantum symmetric pairs.
3) Cluster algebras and quantum groups.
4) Yangians and quantum affine algebras.
5) Geometric representation theory of quantum affine algebras.
The topics of the course will be chosen during an organizational meeting in December 2022 according to the interests of the participants and their background. The interested students are invited to contact the instructor in due time.
The course will be held in hybrid format. Lectures will take place in January-March 2023. 30 hours.
Title and credits: Normal Families, 6 CFU
Teacher: Anna (Miriam) Benini
Syllabus: Consider a family of holomorphic maps from a complex manifold M to a compact complex manifold X. It is natural to ask oneself under which hypothesis such family is precompact in Hol(M,X), i.e. every sequence has a converging subsequence. A precompact family is called normal. When M=\C the complex plane, and X is the Riemann sphere, there is a famous criterion by Montel which claims that precompactness follows if the family omits 3 points in the Riemann sphere.
The theory of normal families plays a crucial role in holomorphic iteration and more generally in functional analysis. We will study normal families of holomorphic/meromorhpic maps from \C to the sphere and also give some informations on normal families in higher dimensions.
Bibliography:
1) Schiff, Joel L.
Normal families.
Springer-Verlag, New York, 1993. xii+236 pp. ISBN: 0-387-97967-0
2) Lyubich, Mikhail
Conformal Geometry and Dynamics of Quadratic Polynomials, vol I-II
Manuscript
3) Additional references for several complex variables to be decided later
Format: Individual Reading + Presentations + Exercises + 12 hours
Dates: November 2022 - January 2023
Title and Credits: Fourier and Laplace transforms and some applications, 4 CFU
Teacher: Marzia Bisi
Syllabus: Fourier transform: from Fourier series to Fourier transform, definition of inverse transform, transformation properties, convolution theorem, explicit computation of some transforms, applications to ODEs and PDEs of some physical problems. Laplace transform: definition, region of convergence, transformation properties, Laplace transform of Gaussian distribution, applications to some Cauchy problems. Definite integrals by means of residue theorem: integrals of real functions, and integrals of Fourier and Laplace useful to evaluate inverse transforms; theorems (with proofs) and examples.
Dates 2022/2023: reading course; pdf slides and videos of all lectures are available on-line, number of expected hours: 24 + individual project.
Title and Credits: Extended kinetic theory and recent applications, 4 CFU
Teachers: Marzia Bisi, Maria Groppi
Syllabus: The course is intended to provide an introduction to classical kinetic Boltzmann approach to rarefied gas dynamics, and some recent advances including the generalization of kinetic models to reactive gas mixtures and to socio-economic problems.
Possible list of topics:
- distribution function and Boltzmann equation for a single gas: collision operator, collision invariants, Maxwellian equilibrium distributions;
- entropy functionals and second law of thermodynamics;
- hydrodynamic limit, Euler and Navier-Stokes equations;
- kinetic theory for gas mixtures: extended Boltzmann equations and BGK models;
- kinetic models for reacting and/or polyatomic particles;
- Boltzmann and Fokker-Planck equations for socio-economic phenomena, as wealth distribution or opinion formation.
Bibliography:
- C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.
- M. Bisi, M. Groppi, G. Spiga, Kinetic Modelling of Bimolecular Chemical Reactions, in “Kinetic Methods for Nonconservative and Reacting Systems” edited by G. Toscani, Quaderni di Matematica 16, Dip. di Matematica, Seconda Università di Napoli, Aracne Editrice, Roma, 2005.
- L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014.
Dates 2022/2023: About 18 hours in January - February 2023 (flexible). The interested Ph.D. students are asked to contact the teachers in advance to define the calendar.
Title and Credits: Numerical methods for option pricing, 2 CFU (50 ore)
Teacher: Chiara Guardasoni
Syllabus:
- Introduction to differential model problems for option pricing in the Black-Scholes framework
- Analysis of peculiar troubles and advantages in application of standard numerical methods for partial differential problems: Finite Difference Methods, Finite Element Methods, Boundary Element Method
Dates 2022/2023: reading course always available.
Title and Credits: Semigroups of bounded linear operators and applications to PDEs, 6 CFU.
Teacher: Luca Lorenzi
Syllabus: In this course we present the theory of semigroups of bounded operators in Banach spaces, paying particolar attention to analytic semigroups. Applications are given to the analysis of partial differential equations of parabolic type.
February-May 2023.
Title and Credits: Introduction to Geometric Measure Theory, 6 CFU
Teacher: Massimiliano Morini
Syllabus: The course covers the following topics: review and complements of Measure Theory; covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems; rectifiable sets and rectifiability criteria; the theory of sets of finite perimeter; applications to geometric variational problems; the isoperimetric problem; the partial regularity theory for quasi-minimiser of the perimeter.
Hand-written notes of the whole course are available in Italian on the Elly platform.
Further references:
- L.C Evans and R.F. Gariepy: "Measure Theory and Fine Properties of Functions"
- F. Maggi: "Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory"
Dates 2022/2023: reading course.
Title and Credits: Several complex variables, 6CFU
Teacher: Alberto Saracco
Syllabus: Theory of several complex variables. Hartogs theorem, Cartan-Thullen theorem, Kontinuitatsatz. Domains of holomorphy, Levi convexity and plurisubharmonic functions. Cauchy-Riemann equation. Sheaves and cohomology (Cech cohomology). The course will be mainly based on Chapters 1-6 of the book by Giuseppe Della Sala, Alberto Saracco, Alexandru Simioniuc and Giuseppe Tomassini: "Lectures on complex analysis and analytic geometry", Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] 3, Edizioni della Normale, Pisa (2006).
Dates 2022/2023: reading course.
Title and Credits: Basic theory of the Riemann zeta-function, 3 CFU
Teacher: Alessandro Zaccagnini
Syllabus: Elementary results on prime numbers. The Riemann zeta-function and its basic properties: analytic continuation, functional equation, Euler product and connection with prime numbers, the Riemann-von Mangoldt formula, the explicit formula, the Prime Number Theorem. Prime numbers in all and "almost all" short intetvals.
Dates 2022/2023: late Winter, early Spring 2023.
Title and Credits: Decision theory for artificial intelligence, 6 CFU
Teachers: Federico Bergenti
Syllabus: The course introduces students to the topics of Decision Theory that are relevant for Artificial Intelligence. In particular, the couse discusses decision-theoretic planning and learning through the following agenda: brief review of random variables and stochastic processes (if needed), discrete-time Markov chains, Markov decision processes, base algorithms for automated planning using Markov decision processes (e.g., value iteration and policy iteration), base algorithms for machine learning using Markov decision processes (e.g., Q-learning and SARSA), brief overview of additional topics (e.g., partially-observable Markov decision processes, game-theoretic planning). The course is delivered as a set of classes and exercize sessions tailored to the specific needs of attending students.
Dates 2022/2023: Two weeks in March or April 2023.
