## Parma

**Lectures take place at “Dipartimento di Scienze Matematiche, Fisiche e Informatiche”, Parco Area delle Scienze 53/A, 43124 Parma.**

Title: **Numerical methods for boundary integral equations**

Teacher: Prof. Aimi Alessandra

Syllabus: The course is principally focused on Boundary Element Method (BEM).

Lectures involve: Boundary integral formulation of elliptical 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 requested.

References will be provided directly during the course.

Dates: Lectures will take place in Spring 2019 at the University of Parma for an amount of 24 hours.

Precise dates will be decided together with the interested Ph.D. students, who are encouraged to contact the teacher in advance.

Title: **Constraint Satisfaction Problems**

Teacher: Bergenti Federico, Monica Stefania

Syllabus: The course is intended to provide an introduction to the current research on Constraint Satisfaction Problems (CSPs) to students with no specific background in Computer Science or Artificial Intelligence. The course starts with an introduction to CSPs and with an overview of algorithms for constraint satisfaction based on heuristic search. Then, algorithms for constraint propagation are presented (e.g., arc consistency and hyper-arc consistency), and the forms of consistency that they achieve are discussed. Finally, algorithms to treat polynomial constraints are shown (e.g., Buchberger's algorithm and cylindrical algebraic decomposition), with emphasis on polynomial constraints over finite domains (e.g., based on Bernstein polynomials and on Rivlin's bound)

Dates: 8-10 hours in November-December 2018 (flexible).

Title: **Fourier and Laplace transforms and some applications**

Teacher: Bisi Marzia

- 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.

Title: **Extended kinetic theory and recent applications**

Teachers: Bisi Marzia, Groppi Maria

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.

Title: **Lectures on selected topics in Biomathematics**

Teachers: Groppi Maria (Parma)

Syllabus: Some differential models in biomathematics will be introduced and studied in this series of lectures. Attention will be paid both to the construction of the models and to the study of the resulting mathematical problems, in the context of (ordinary-partial-integro) differential equations.

A possible list of topics (to be better defined with the audience):

- Mathematical models in electrocardiology and neural network models
- Optimal control problems in epidemiological systems
- Bifurcations in predator prey systems
- Kinetic models for tumor-immune system dynamics
- Mathematical models in landscape ecology
- Turing instability and pattern formation in reaction diffusion system

Title: **Infinite Dimensional Analysis**

Teachers: Lunardi Alessandra (Parma), Miranda Michele (Ferrara)

Syllabus: This is an introductory course about analysis in abstract Wiener spaces, namely separable Banach or Hilbert spaces endowed with a nondegenerate Gaussian measure. 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 Poincare' 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). In addition, lecture notes prepared by the teachers will be available.

Dates: November 2018/ February 2019 (approx. 40 hours, either in Parma or in Ferrara according to the needs of the students)

Title: **Internet Seminar, on Ergodic Theory **

Virtual teacher: Tanja Eisner (Leipzig), Balint Farkas (Wuppertal)

Local teacher: Lunardi Alessandra

Syllabus: We present some of the classical and more modern ergodic theorems and discuss connections to other areas of mathematics such as Fourier analysis and number theory.

The course is suitable for bachelor (about finishing BSc) and master students with a firm background in functional analysis and measure theory. Interested PhD students might find it inspiring for their own research.

Course structure and dates:

Course phase : October 2018-Februrary 2019

Weekly online lecturers (approx. 13-15).

Project phase : March-June 2019

Registered participants may apply for projects offered by the virtual lecturers and invited coordinators.

Successful applicants will work on a project in an international team of 3-4 and prepare a talk of 60-120 minutes (allocated to the entire team) based on a research paper.

Workshop: 10-16 June 2019, Wuppertal

The prepared talks are to be presented at the final workshop. More information on the workshop will follow in due course.

It is possible to take part in the first phase only. Phases 2 and 3 require participation in the previous phases.

Title: **Inverse problems**

Teachers: Yamamoto Masahiro (The University of Tokyo)

Syllabus:

- Basics of the one-dimensional wave equations: energy estimate, the multiplier method and other fundamental estimates.
- An introduction to inverse problems: proof of the observability inequality and the stability for a simple inverse source problem for the one-dimensional wave equation.
- Simplified derivation of Carleman estimates for the wave equation and the heat equation in the one-dimensional case.
- Carleman estimates in the general case of hyperbolic equations.
- Applications of Carleman estimates to inverse source and coefficient problems: a general methodology for proving the Lipschitz stability in determining a spatially varying source and/or a coefficient by lateral boundary extra data.
- Inverse source/coefficient problems for parabolic equations.
- Inverse source problems for other kinds of partial differential equations: integrodifferential equations, Maxwell's equations, Schrödinger equation, elasticity equations.

M. Bellassoued and M. Yamamoto, Carleman "Estimates and Applications to Inverse Problems for Hyperbolic Systems", Springer-Japan, Tokyo, 2017

Dates: April-May 2019 (contact Luca Lorenzi if you are interested).

Title: **Advanced stochastic processes (reading course)**

Teacher: Morandin Francesco

Syllabus:

- Review of basic probabilistic concepts.
- Large deviations theory

- Introduction to large deviations; Calculus of large deviations.

- Cramer's theorem, Gartner-Ellis theorem, Sanov's theorem. - Levy processes

- Intro and basic properties of Brownian motion; Reflection principle, quadratic variation.

- Concentration inequality for martingales; Applications to the theory of random graphs.

- Poisson point processes; Lévy processes; subordinators.

- Stable distributions.

- Continuous-time Markov chains. - Weak convergence theory and applications.

- Probability on metric spaces; Weak convergence of probability measures; Portmentau theorem.

- Construction of a Brownian motion; Functional Central Limit Theorem.

- Stable Central Limit Theorem.

- Convergence of empirical processes.

Billingsley - Weak Convergence of Measures

Protter - Stochastic integration and differential equations

Dembo, Zeitouni - Large Deviations Techniques and Applications

Dates: 5 November 2018 - 29 March 2019

Title: **Shell models of turbulence**

Teacher: Morandin Francesco

- Navier-Stokes and Euler equations in Fourier components
- Turbulence, energy cascade and (lack of) regularity
- Shell models of turbulence and phenomenology
- GOY, sabra and numerical results
- Katz-Pavlovic and dyadic models
- Stochastic models
- Tao model and the realistic shell model
- Intermittent tree dyadic model
- Multifractal formalism

Teacher: Saracco Alberto

**One course out of the following three:**

- Title:
**Algebras of functions**

Abstract: An introduction to the theory of Banach and Fréchet algebras, both from the abstract point of view and from the concrete point of view of algebras of holomorphic functions. - Title:
**Convexity in C^n**

Abstract: Abstract: A survey of the various notions of convexity in C^n (convexity, pseudoconvexity, holomorphic convexity, C-convexity) and their applications in complex geometry. - Title:
**Complex distances and hyperbolicity**Abstract: A survey of the various notions of distance in complex manifolds, paying attention to various notions of hyperbolicity. - Title:
**Cohomology and Vector Bundles**Abstract: An introduction to sheaves and Dolbeault cohomology and its application to various extension problems in the theory of several complex variables.

Corsi della LM appropriati anche per studenti di dottorato

- Calcolo delle Variazioni (docente M. Morini)
- Modelli Matematici della Finanza (docenti M. Bisi e C. Guardasoni)
- Equazioni di Evoluzione (docente A. Lunardi)
- Teoria Cinetica (docente G. Spiga)