Latest research directions in mathematics

During the course we will listen to talks of prominent mathematicians who will talk about their research. The goal is to see in which direction the mathematics and its applications are heading nowadays.

We will be meeting on Tuesdays starting at 13:15 in A.2.22@C19.

1. 03.10: Karolina Kropielnicka (IMPAN)- Hydrogen atom interacting with light as a meeting point of quantum physics, numerical methods and functional analysis. The taste of applied and computational mathematics

• If mathematics is the language describing the world, differential equations form a large part of its vocabulary. The significance of differential equations, as a natural, mathematical tool in describing phenomena of our everyday lives does not need to be explained or substantiated. Once a differential system is designed, there is an inevitable necessity of finding its solutions, and here numerical analysis plays a central role. Computational partial differential equations are arguably the largest and fastest evolving area of numerical mathematics. The variety, importance and dynamics of advances in PDEs explains the rapid pace and wide scope of development of numerical tools dedicated to various particular problems in the rich area of differential equations. The interaction of hydrogen atom with light is modelled by singular Schrodinger equation and can be predicted resorting to numerical approximation. In this lecture we will understand why it is a challenging task and why standard numerical approaches do not work. Basing on the “simple” example I will show the place where applications meet theory and why they are equally important and cannot exist in two different worlds.

1. 10.10: Tomasz Jakubowski (PWr) - Fractional Laplacian in differential equations and probability
2. 17.10: María López Fernández (Universidad de Málaga) - Contour integral methods or Laplace transform based methods for evolution equations

• The solution to certain important evolution equations can be efficiently approximated in time by applying the numerical inversion of the Laplace transform. This approach is a competetive alternative to more traditional time stepping methods, since it allows a high level of parallelism, it does not require to compute the history and its accuracy is not affected by the regularity in space of the initial data. The inverse Laplace transform provides a suitable contour integral representation of the solution in the complex plane, for which exponentially convergent quadratures can be devised. In this lecture we will review the basic notions of this approach.

1. 26.10: Rene Schilling (Technische Universitat Dresden) - Room A.1.14@C19, 11:15 - Maximal Inequalities in Probability Theory
2. 31.10: Juan Rocha Martin (Universidad de Las Palmas de Gran Canaria) - Maximum principle for time-fractional diffusion equations with the Caputo-Katugampola derivative - Slides
3. 07.11: Florian Fischer (Potsdam) - p-Parabolic Graphs

• By the celebrated uniformisation theorem of Klein, Koebe and Poincaré in the linear (p = 2)-case, any simply connected Riemann surface is conformally equivalent to either the sphere (surface of elliptic type), the Euclidean plane (surface of parabolic type) or the hyperbolic plane (surface of hyperbolic type). Since the sphere is the only one with compact surface, the type problem is to decide whether a surface is of hyperbolic or of parabolic type. It could be shown more than 50 years ago that the recurrence of Brownian motion is equivalent to the (p=2)-parabolicity of the Riemann surface. Since then many geometry, analytic and probabilistic characterisations of parabolicity have been found. In this talk, I will state some of them in a quasi-linear setting on infinite graphs. A goal is to state and prove a Kelvin-Nevanlinna-Royden-type characterisation. The talk shows work in progress with Andrea Adriani (U. dell'Insubria) and Alberto Setti (U. dell'Insubria).

1. 14.11: Class moved to 12.12
2. 21.11: Agnieszka Wyłomańska (PWr) - Fractional Brownian motion with constant and random Hurst exponent - mathematical background and statistical testing
3. 28.11: Łukasz Płociniczak (PWr) - Numerical methods for nonlinear and nonlocal PDEs
4. 05.12: PhD students' presentations
5. 12.12: Bartosz Trojan (PWr) ¡Room change!: A.4.1 - Compactifications of affine buildings
   • In this talk I am going to introduce affine buildings and indicate what are the analytic problems one can be interested in. In particular, I describe several compactification procedures and motive the study of random walks. I am going to formulate what is Martin compactification and explain why it is important.
6. 19.12: Karol Szczypkowski (PWr) - Stable processes with partial resetting
   • We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a Lévy process $\mathbf{Y}$ by partial resetting at Poisson moments. We focus on $\mathbf{Y}$ being a strictly $\alpha$-stable process with $\alpha\in (0,2]$ having the density. Among other things: We analyze properties of the transition density $p$ of the process $\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $\rho_{\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\mathbf{Y}$.
7. 09.01: Krzysztof Burnecki (PWr) - Insurance-linked securities tied to natural catastrophes
8. 16.01: Katarzyna Pietruska-Pałuba (UW) - Hardy inequality for the fractional Laplacian on Lp
   • We will describe Hardy inequalities for the fractional Laplacian in Lp, understood as the generator of the stable process. These inequalities involve a counterpart of the classical Dirichlet form of the underlying process (denoted Ep). Time-permitting, we will also present the Hardy-Stein identity in Lp: it is concerned with norms of functions in Lp and uses forms Ep as well.
9. 23.01: PhD students' presentations
10. 30.01: PhD students' presentations. Conclusion