### 30.01.2023 15:00 Giulio Zucal (MPI): Action convergence: from graph to hypergraph limits

The theory of graph limits considers the convergence of sequences of graphs with a diverging number of vertices. From an applied perspective, it aims to represent very large networks conveniently. Until recently, however, particular cases for graph limits have been investigated separately, while hypergraph limits are even less well-developed. In this talk I will give a brief introduction to action convergence, a recent unified approach to graph limits based on functional analysis and measure theory. Moreover, I will present some work in progress on the extension of action convergence to hypergraphs.

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### 30.01.2023 15:45 Parvaneh Joharinad (MPI): Curvature and hyperconvexity

The aim of this presentation is to briefly present generalized (sectional) curvature and see which kind of relations between data points it evaluates and what kind of information is revealed through this quantity.
While in topological data analysis the objective is to extract qualitative features, the shape of data, geometric data analysis mainly deals with quantitative features of data. For instance, the prominent scheme of manifold learning is applied to find the comparatively low dimensional Riemannian manifold on which the data set fits best. It then raises the question of whether one can anticipate some geometric properties from initial model before finding this manifold structure.
The most important quantitative measures that in a good extent reveal the geometry of a Riemannian manifold are its (sectional) curvatures. Therefore, we wish to see how one can determine the curvature of data and how does it help to derive the salient structural features of a data set.

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### 30.01.2023 16:30 Peter K. Friz (TU Berlin): Martingales and rough paths

Despite being very different in nature, martingales and rough paths have many similarities and their interplay is most fruitful. As a concrete example, I will introduce the recent notion of rough stochastic differential equations and explain its importance in filtering, pathwise control theory and option pricing under (possibly rough) stochastic volatility. (Joint work with numerous people, including Pavel Zorin-Kranich, Khoa Lê, Antoine Hocquet, Peter Bank, Christian Bayer and Luca Pelizzari.)

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### 31.01.2023 16:00 Mia Runge: Geometric Inequalities Involving Different Diameter Definitions

When considering non-symmetric gauges there are several ways to define the diameter of a convex body. These correspond to different symmetrizations of the gauge, i.e., means of the gauge $C$ and $-C$. We study inequalities involving the inradius, circumradius and diameter and present examples and results that confirm that not only does studying the symmetrizations help to understand the diameters better but also the other way around.

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### 31.01.2023 16:00 Florian Grundbacher: Ellipsoids in convex bodies of a given asymmetry

In „Ellipsoids of maximal volume in convex bodies“ Keith Ball proved a general bound on the volume of k-dimensional ellipsoids in n-dimensional convex bodies in relation to their John ellipsoid. A stronger bound is known in the symmetric case. Our goal was to connect these results by establishing a bound depending on the John asymmetry s_0. We could prove a tight bound for all k and all asymmetry values s_0 not in (1,1+2/n), and characterize the equality cases.

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