Parma 2023/24

Title and Credits: Numerical approximation of contact and friction problems, 6 CFU

Teacher: Franz Chouly (Université de Bourgogne, INdAM visiting)

Syllabus:

1-Introduction

2-Fractional Sobolev spaces

3-Signorini problem and a naïve finite element approximation

4-New approximation techniques and optimal error bounds

5-Extension to friction (Tresca, Coulomb)

6-Extension to elastodynamics and impact of elastic bodies

Bibliography

Chouly, P. Hild, and Y. Renard, Finite element approximation of contact and friction in elasticity, no. 48 inAdvances in Continuum Mechanics, Birkhäuser, Springer, 2023. ISBN 978-3-031-31422-3

Dates 7-9-14-16-21-23-28-30 november 2023: 14:30-16:30;  16h lectures +  home assignment; the students who are interested to follow lectures online, are asked to contact in advance alessandra.aimi@unipr.it

Title and Credits: Morse Theory 4 CFU

Teacher: Leonardo Biliotti

Syllabus:non-degenerate smooth functions on manifold, homotopy thype  in terms of critical values, morse inequalies, the existence of non-degenerate functions, the Lefschets Theorem on Hyperplena sections, the calculus of variations applied to geodesic.

References: ''Morse Theory '', Milnor, ''An invitation to Morse Theory'', Nicolaescu

Dates 2023/2024:  reading course.

__________________________________________________________________________________________________________

Title and Credits: Complex surfaces and their automorphisms 5 CFU

Teacher: Andrea Cattaneo

Syllabus: the aim of the course is to introduce the main techniques used in complex and algebraic geometry to study the geometry of compact complex surfaces. We will review the main results of the theory: the Riemann—Roch theorem with its implications, the effects of a blow up on the topology and the cohomology of a surface and Castelnuovo’s contraction theorem, the concept of minimal surface and the problem of classification (Castelnuovo’s rationality criterion and Enriques—Kodaira classification). Finally, we will focus on automorphisms of surfaces, focussing on what is known about their automorphism groups in particular for surfaces of Kodaira dimension 0.

Dates: 8 weeks, starting mid January 2024

____________________________________________________________________________________________________________

Title and credits: Topics in Representation Theory 6 CFU

Teacher: Andrea Appel

__________________________________________________________________________________________________________-

Title and Credits: Normal Families Format 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 a 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 information 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

Dates will be decided with students; Individual Reading + Presentations + Exercises + 12 hours lectures.

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 Lectures will take place in II semester 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: 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: February-March 2024 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, 6 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 February 2024; the interested  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 2023/24: reading course always available.

____________________________________________________________________________________________________________

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 2023/2024;: the interested  students are asked to contact the teachers in advance to define the calendar.

___________________________________________________________________________________________________________

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.

Dates: February-May 2024.

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 2023/2024: 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 2023/2024:  reading course.

____________________________________________________________________________________________________________

Title and Credits: Introduction to Elliptic partial differential equations 6 CFU

Teacher: Paolo Baroni

Syllabus: the course aims at introducing basic problems and classic existence and regularity techniques for uniformly elliptic PDEs with linear growth. In particular, the following topics will be analyzed, more or less in detail according to the students’ interest:
- Harmonic functions, weak formulation and Weyl's Lemma.
- Second order regularity for the Poisson equation via Calderon-Zygmund decomposition and singular integrals.
- Second order Sobolev regularity for equations with constant coefficients.
- Linear equations with variable coefficients: W^{1,q} estimates for continuous coefficients.
- Campanato spaces and Schauder theory for linear equations with Holder coefficients and data.
- De Giorgi theory (Holder regularity of solutions for measurable coefficients).
- Harnack inequalities, expansion of positivity for equations with measurable coefficients.
- Gehring theory (higher integrability of the gradient).

Dates: The course is intended to span approximatively two consecutive months (to be decided) between October 2023 and February 2024. Students are encouraged to contact the lecturer as soon as possible in order to fix a timetable.