In this talk we are going to study branching processes and its extension when the offspring distribution is varying over time. We are going to analyze its extinction probability. By using a two-spine decomposition technique we are going to give the law of the process conditioned on non extinction.

Zakhar Kabluchko (University of Münster)
Random beta tessellations

We shall define and study several families of random tessellations of the Euclidean space, the sphere and the hyperbolic space that generalize the classical Poisson-Voronoi and Poisson-Delaunay tessellations. The talk is based on a joint work with Anna Gusakova and Christoph Thäle.

Lukas Gonon (LMU, Munich)
Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

We study the expression rates of deep neural networks for option prices written on high-dimensional baskets of risky assets, whose log-​returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process that ensure that the size of the DNN required to achieve a given approximation accuracy grows only polynomially with respect to the dimension of the Lévy process and the reciprocal of the approximation accuracy, thereby overcoming the curse of dimensionality and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.
In addition, we exploit parabolic smoothing of Kolmogorov partial integrodifferential equations for certain multivariate Lévy processes to present alternative architectures of ReLU DNNs that provide higher approximation rates, however, with constants potentially growing exponentially with respect to the dimension. Under stronger, dimension-​uniform non-​degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the present results.
The talk is based on joint work with Christoph Schwab.

Lisa Hartung (University of Mainz)
Branching Brownian motion among obstacles

In this informal talk, after introducing the class of models I will give an overview of the existent results in the literature on the behaviour of branching Brownian motion among obstacles. These results mainly concern the survival probability in the case of hard obstacles and the population size in the case of mild obstacles.
Then I will explain a couple of open questions which I would like to get a better understanding of.

Jiequn Han (Princeton University)
Deep BSDE Method and its Convergence Analysis for High-Dimensional PDEs & Games

Developing algorithms for solving high-dimensional partial differential equations, controls, and games has been an exceedingly difficult task for a long time, due to the notorious "curse of dimensionality". In the first part of the talk, I will introduce the Deep BSDE method for solving high-dimensional parabolic PDEs. The algorithm builds on the reformulation of backward stochastic differential equations and utilizes deep neural networks as efficient approximators to unknown high-dimensional components. Numerical results of various examples, including multi-agent games, demonstrate the efficiency and accuracy of the proposed algorithms in high-dimensions. In the second part of the talk, I will introduce some convergence analysis of the Deep BSDE method in terms of a posteriori error estimation and an upper bound for the minimized objective function.

Arno Siri Jégousse (UNAM, Mexico)
Site frequency spectrum of the Bolthausen-Sznitman coalescent

In this talk, I will study the concept of site frequency spectrum (SFS), which happens to be one of the most relevant statistics in population genetics. It is of particular use for genealogical model selection and estimation in evolution models. The SFS is closely related to the shape of the genealogical tree of the observed sample of a population.
In the particular case of the Bolthausen-Sznitman coalescent, which is now accepted to be the null model for rapidly evolving populations or populations under strong selection, the limit behaviour of the SFS can be studied thanks to approximations with random walks. More interestingly, there exists a construction of the coalescent by means of random recursive trees which yields very precise approximations of the moments of the SFS. This technique also gives new asymptotic results, leading to a complete picture of the statistic.
This is a joint work with Götz Kersting (Frankfurt) and Alejandro Wences (Mexico City).

Eva Kopfer (University of Bonn)
Random Riemannian Geometry

We study random perturbations of Riemannian manifolds by means of so-called Fractional Gaussian Field. The fields act on the manifolds via conformal transformation. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline.

We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will change under the influence of the noise.

Gilles Bonnet (University of Bochum)
Concentration and Cumulants for Stabilizing Functionals on Point Processes

In this talk I will present aspects of the project "Concentration and Cumulants for Stabilizing Functionals on Point Processes". In particular, the notion of stabilizing functional will be introduced and several classical methods to obtain concentration bounds presented.

Christian Beck (WW University Münster)
Multilevel Picard approximations for semilinear elliptic PDEs

Multilevel Picard approximation methods have been successfully used to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic PDEs (see, e.g., Hutzenthaler et al. https://royalsocietypublishing.org/doi/full/10.1098/rspa.2019.0630). In this talk we extend multilevel Picard approximations for semilinear elliptic PDEs, state that they can overcome the curse of dimensionality in the numerical approximation of semilinear elliptic PDEs, point out differences between the parabolic and the elliptic situation, and comment on possible directions of future research.

Alexander Drewitz and Bingxiao Liu (University of Cologne)
Random polynomials and random Kähler geometry

We are presenting our SPP project which focuses on the interplay between complex geometry and probability theory. More precisely, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques in order to study several problems concerning local and global statistical properties of zeros of holomorphic sections of holomorphic line bundles over Kähler manifolds. A particularly important instance of this setting is given by the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial / sections ensembles, universality of their distributions, central limit theorems and large deviation principles in this context.

Felix Lindner (University of Kassel)
On a class of stochastic differential-algebraic equations arising in industrial mathematics

The dynamics of inextensible fibers in turbulent airflows is of interest, e.g, in the context of mathematical models for spunbond production processes of non-woven textiles. In this talk, a model based on higher-index stochastic differential-algebraic equations is presented. It involves an implicitly given Lagrange multiplier process, the explicit representation of which leads to an underlying stochastic ordinary differential equation with non-globally monotone coefficients. Strong convergence is established for a half-explicit drift-truncated Euler scheme which fulfills the algebraic constraint exactly.