Courses in Parma
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 2021/2022: Lectures will take place in Spring 2022 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: Introduction to quantum groups, 4CFU
Teacher: Andrea Appel
Syllabus: The course will be a blend of mostly representation theory (quantum groups and Hopf algebras), a bit of basic complex algebraic geometry (blowups), some category theory (braided monoidal categories), and some deformation theory (Hochschild cohomology). The course is intended for a general mathematical audience: I will do everything from scratch, assuming only the basic notions in algebra and geometry.
The first part of the course will provide a parallel between the classical theory of the Lie algebra sl(2) and that of its quantum counterpart Uqsl(2), with a special focus on the role of the universal R-matrix and the Yang-Baxter equation.
The topics discussed in the second part of the course will depend upon the main interests of the audience. Potential topics are: monodromy of the Knizhnick-Zamoldchikov equations and Kohno-Drinfeld theorem; Yangians and quantum loop algebras; Etingof-Kazhdan quantization of Lie bialgebras; categorification and Khovanov-Lauda-Rouquier algebras; Reshetikhin-Turaev invariants; quantum groups at root of unity.
More information here
Dates 2021/22: 24 hours, in the period January-March 2022
Title and credits: Introduction to complex dynamics, 3 CFU
Teacher: Anna (Miriam) Benini
Syllabus: We will give an introduction to complex dynamics following Milnor's book and notes by M. Lyubich. We will investigate the dynamics of polynomials, rational maps and transcendental maps.
Dates: January 10-February 24th, 2022
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 2021/2022: reading course; pdf slides and videos of all lectures are available on-line, number of expected hours: 24 + individual project.
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.
- 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 2021/2022: About 18 hours in January - February 2022 (flexible). The interested Ph.D. students are asked to contact the teachers in advance to define the calendar.
Title and Credits: Interface and Contact Problems, 3 CFU
Teacher: Heiko Gimperlein, Heriot–Watt University, UK (Visiting Professor)
- Modelling of interface problems between materials: interface conditions and friction laws,
- Obstacle problems, friction and contact: (free) time-independent boundary problems as constrained on nonsmooth variational problems, solution using Uzawa and semismooth Newton methods,
- Nonlinear analysis of variational inequalities: functional analytic background, classical theorem on wellposedness and abstract approximation, regularity of solutions,
- Approximation by finite and boundary elements - from basics to current research: BEM for dummies, variational inequalities and penalty formulations, error analysis, adaptivity, advanced approximation methods, maybe coupling of FEM and BEM,
- Towards time-dependent variational inequalities and dynamic contact.
Dates 2021/2022: May 2 - 13, 2022
Title and Credits: Numerical methods for option pricing, 2 CFU (50 ore)
Teacher: Chiara Guardasoni
- 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 2021/2022: reading course always available
Title and Credits: Spectral theory for operators in Banach spaces and applications to semigroups, 6 CFU
Teacher: Luca Lorenzi
Syllabus: The program contains the topics listed here below plus possibly additional material based on the students' interests. To make the course self-contained, an introduction to the theory of semigroups of bounded operators will be provided to students who are not acquainted with this theory.
- A survey on operator theory
1.1 Closed operators. Definitions and different characterizations. Closable operators and closure of an operator.
1.2 Spectrum and resolvent of a (bounded or closed) operator. Basic properties. Spectral radius.
1.3 Normal and selfadjoint operators and their basic properties.
- Compact operators and Riesz-Schauder theory
2.1 Compact operators: definitions, examples, Schauder theorem.
2.2 Riesz-Schauder theory for compact operators.
2.3 Spectral decomposition theorem for selfadjoint compact operators.
- Spectral representation theorem for bounded operators
3.1 Spectral representation theorem for bounded and selfadjoint operators on a separable Hilbert space H.
3.2 Spectral theorem for normal operators
- Spectral representation theorem for unbounded operators
4.1. Adjoint of an unbounded operator, selfadjoint and symmetric operators. Definitions and basic properties.
4.2. Dissipative operators. Definitions and main properties.
4.3. Representation theorem for unbounded selfadjoint operators.
4.4. Spectral mapping theorem for the resolvent operator of closed linear operator.
4.5 Spectral representation theorem for selfadjoint operators.
4.6 Positive operators and minimax theorems for their eigenvalues.
- Spectral mapping theorems for semigroups
5.1 The spectral mapping theorem for the point and residual spectrum.
5.2 Spectral mapping theorem for eventually norm continuous semigroups and consequences.
5.3 Examples and counterexamples.
Dates 2021/2022: from march to the end of may.
Title and Credits: Operator semigroups and evolution equations, 6CFU.
Teacher: Alessandra Lunardi
Syllabus: The course deals with the basic theory of semigroups of linear operators and evolution equations in Banach spaces. Main topics are
- basic spectral theory for linear operators in Banach spaces;
- strongly continuous semigroups and the Hille-Yosida theorem;
- analytic semigroups;
- Cauchy problems for linear evolution equations in Banach spaces: regularity and asymptotic behavior.
Lecture notes (in italian) will be provided. Further bibliography include
- K. Engel, R. Nagel: One-parameter Semigroups for Linear Evolution Equations, Spinger Verlag, Berlin, 1999.
- K. Engel, R. Nagel: A Short Course on Operator Semigroups, Spinger Verlag, Berlin, 2006.
- A. Lunardi: Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser Verlag 1995, 2nd edition 2013.
Dates 2021/2022: reading course.
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.
- 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 2021/2022: reading course.
Title and Credits: Instability and Bifurcation, 6 CFU
Teacher: Paolo Piccione(*), Universidade de Sao Paulo, Brasil
Syllabus: We give an overview of classical results in variational Bifurcation Theory and some geometrical applications, including multiplicity results for Geodesics, Constant Mean Curvature Surfaces, and the Yamabe problem. The detailed program is here.
Dates 2021/2022: november - december 2021.
(*) Supported by INdAM
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 2021/2022: 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 2021/2022: late winter, early spring 2022.
Title and Credits: Holomorphic and isometric immersions, 3CFU
Teacher: Michela Zedda
Syllabus: After an introduction of basic instruments of Kaehler geometry, the course focuses on existence, extension and rigidity results for germs of holomorphic isometries between Kaehler manifolds. In particular, we will introduce Calabi's diastasis function and Calabi's criterium and we will discuss the case of complex space forms and give some explicit examples of holomorphic isometry.
Dates 2021/2022: 24 hours, December 2021-January 2022 (flexible).