Mathematical Colloquium

We present the multilevel Picard approximation method for high-dimensional semilinear parabolic PDEs. A key idea of our method is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of the proposed method grows polynomially both in the dimension and in the reciprocal of the required accuracy. Moreover, we present further applications of the multilevel Picard approximation method and illustrate its efficiency by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe von Wurstemberger.

Empirical process theory is a concept in modern statistics to quantify the deviation of a mean from its expectation uniformly over a class of functions. It has many applications in the statistical analysis of machine learning algorithms and, in particular, minimum risk estimation.
While the theory is well-developed for means based on independent observations, the sitution is much less systematic for dependent data. Most concepts are based on a quantification of dependence via so-called 'mixing'. The most complete theory was developed for beta-mixing, but also results for phi-mixing or Markov chains do exist.
However, mixing coefficients are hard to upper bound even for simple time series models like linear processes and comes with strong conditions. Therefore, we establish a new empirical process theory which is based on quantifying dependence with the so called functional dependence measure. The main advantage is that this measure is easy to upper bound for a large variety of well-known time series models.
The talk starts with a short recapitulation of some results of empirical process theory for independent variables and beta-mixing sequences. Then, the functional dependence measure is introduced and the corresponding empirical process theory results are presented.

The number of integral points of the integral inflations of a lattice polytope in $\mathbb{Z}^n$ is given by the polytope's Ehrhart polynomial, a well-studied invariant in polyhedral geometry. How is this invariant distributed over all superlattices of $\mathbb{Z}^n$? I will link this question with classical generating functions arising in number theory, viz. Hecke series, Bruhat-Tits buildings and Hecke eigenfunctions. This is joint work with Claudia Alfes and Joshua Maglione.