# Department Colloquium Summer 2022

## Talks in the Summer Semester 2022

At the Colloquium of the Department of Mathematics international researchers report on their work. The colloquium will be held on Wednesdays in "Hörsaal 3" (MI 00.06.011). We invite all interested parties.

### Dates and topics at the Department Colloquium

#### Structure-preserving finite elements and particles for plasma simulation

27 July 2022, 14:30 - 15:30

**PD Martin Campos Pinto**

Max-Planck-Institut für Plasmaphysik - IPP

Despite steady progresses over the last half century, the numerical simulation of fusion plasmas remains a huge challenge for applied mathematicians and computational physicists, mostly due to the complex nonlinear interactions that occur between multiple physical scales.

The objective of this talk is to present some promising tools and open projects in this field: In a first part I will describe a novel structure-preserving discrete framework that provides stable, high-order and efficient solvers for Maxwell's equations in complex domains. This framework builds upon the Finite Element Exterior Calculus theory which preserves at the discrete level the de Rham geometric structure of the exact problems, but also allows a greater locality in the computations and a greater modularity in the implementation.

In a second part I will show how to couple these generic field solvers with particle approximations of the Vlasov equation describing the evolution of a collisionless plasma, while preserving the Hamiltonian structure of the exact system.

I will conclude with some open problems that are currently tackled in the NMPP division of the Max-Planck Institute for Plasma Physics.

#### Stationary vine copula models for multivariate time series

27 July 2022, 16:00 - 17:00

**Prof. Aleksey Min **

Technical University of Munich

Copulas are multivariate distribution functions with uniformly distributed margins on the unit interval [0,1]. Despite their simplicity, they are very helpful for modeling a dependence structure of multivariate data in applied science. In the talk, I outline a flexible construction of copulas using graphical models vines. Recently, the so-called vine copulas are successfully used to model univariate and multivariate time series. A time series consists of multiple observations indexed by time. Classical time series models allow for only linear dependence between variables and time points. Vine copulas can conveniently capture cross-sectional and temporal dependence of multivariate time series. In the talk, I derive the maximal class of graph structures that guarantee stationarity under a natural and verifiable condition. I also discuss computationally efficient methods for estimation, simulation, and prediction. The theoretical results allow for misspecified models and, even when specialized to the iid case, go beyond what is available in the literature. The talk is based on the joint work with Thomas Nagler and Daniel Krüger.

#### The logarithmic Minkowski problem

13 July 2022, 14:30 - 15:30

**Prof. Martin Henk**

Technical University of Berlin

The classical (discrete) Minkowski problem asks for necessary and sufficient conditions such that a given set of unit vectors \(a_i\) and positive numbers \(\alpha_i\), \(1\leq i\leq m\), are the facet data of a polytope, i.e., there exists a polytope PPP having facets in the directions \(a_i\) of area \(\alpha_i\). This problem was solved by Minkowski, and it is a corner stone of classical Brunn-Minkowski theory. The analogous problem in modern convex geometry and within the \(L_p\)-Brunn-Minkowski-theory is known as the \(L_p\)-Minkowski problem. Of particular interest is the limit case \(p=0\) and the associated so called logarithmic Minkowski problem. Here the problem is to decide when the given data are the cone data of a convex polytope PPP containing the origin, i.e., \(\alpha_i\) is the volume of the cone generated by the origin and the facet in direction \(a_i\).

In the talk we survey on the state of the art of the logarithmic Minkowski problem.

#### Inaugural lecture: Two tales of modelling financial markets - from rough stochastic volatility to deep generative models

13 July 2022, 16:00 - 17:00

**Prof. Blanka Horvath**

Technical University of Munich

Rough fractional stochastic models have been used for several decades to model natural phenomena. In mathematical finance, the family of so-called 'rough volatility models' - where the volatility process has rougher sample paths than Brownian motion - has attracted tremendous interest in the past years, due to its ability to reproduce several key features of

- financial time series [Gatheral-Jaisson-Rosenbaum '18], and
- of the observed skew of implied volatility [Alos-Leon-Vives '07, and Fukasawa '11], as well as
- due to the fact that rough volatility models arise as scaling limit of microstructure models [El Euch-Fukasawa-Rosenbaum `18, Jaisson-Rosenbaun '20].

Since the emergence of Deep Pricing and Hedging the need for models that reliably reproduce key features (stylized facts) of financial markets has become even more pronounced. Deep generative modelling techniques make it possible to model financial time series in a fully flexible data-driven way that is not limited to the choice of a stochastic financial model, but instead where features are encoded through in the rough path signature of the price path.

In this talk we discuss why realistic stochastic models (such as rough volatility) are of paramount importance for such data driven deep algorithms and also highlight some of the most recent challenges that arise for mathematical finance in this new setting.

Further information can be found on the overview page of the Colloquium of the Department of Mathematics.