The research seminar Analysis & Geometry (A&G) joint with EPFL is organised by the research groups of Dr. Julia Hörrmann, Prof. Vladimir Matveev, Prof. Marc Troyanov (EPFL) and Prof. Thomas Wannerer.

The research seminar Dynamical Systems & Mathematical Physics (DS&MP) is organised by the research groups of Prof. David Hasler, Prof. Daniel Lenz and Prof. Tobias Jäger.

**When**: Thursdays @ 14:15 -16:00 and/or 16:15-17:45

**Where**: The event from 14:15-16:00 takes place in seminar room 3517, Ernst-Abbe-Platz 2 or online via Zoom.

The event from 16:15-17:45 will take place in seminar room 131, Carl-Zeiß-Straße 3 or online via Zoom.

Access data for the online event will be provided as needed.

**Mailing list:** If you would like to be added to the distribution list of the Research Semimar, please send a short e-mail here.

If you want to be deleted from the distribution list, please use this address as well.

# Summer term 2022

## June 9, 2022

*14:00 -16:00 (DSMP),*

**Dr. Timo Spindeler** (Bielefeld)

**Eigenmeasures with respect to the Fourer transform**

*Abstract: Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on R^d. In particular, we classify all periodic eigenmeasures on R, which gives an interesting connection with the discrete Fourier transform, as well as all eigenmeasures on R with uniformly discrete support.*

(Contact: D. Lenz)

## June 23, 2022

*14:00 -16:00 (DSMP),*

**Dr. Markus Lange** (Trieste, Italien)

**Quantum Systems at The Brink: Existence of Bound States, Critical Potentials and Dimensionality**

*Abstract: The existence of bound states plays a crucial role for the properties of quantum systems. I will present a necessary and sufficient condition for Schrödinger operators to have a zero energy bound state. In particular I will show that the asymptotic behavior of the potential is the crucial ingredient.The existence and non-existence result complement each other and exhibit a strong dependence on the dimension.This is based on joint work with Dirk Hundertmark and Michal Jex.*

(Contact: D. Hasler)

## June 30, 2022

*14:00 -16:00 (A&G),*

**Prof. Dr. Daniel Hug** (KIT, Karlsruhe)

**Curvature measures and soap bubbles beyond convexity**

*Abstract: *A fundamental result in differential geometry states that if a smooth hypersurface in a Euclidean space encloses a bounded domain and one of its mean curvature functions is constant, then it is a Euclidean sphere. This statement has been referred to as the soap bubble theorem. Major contributions are due to Alexandrov (1958) and Korevaar--Ros (1988).

While the smoothness assumption is seemingly natural at first thought, based on the notion of curvatures measures of convex bodies Schneider (1979) established a characterization of Euclidean spheres among general convex bodies by requiring that one of the curvature measures is proportional to the boundary measure. We describe extensions in two directions: (1) The role of the Euclidean ball is taken by a nice gauge body (Wulff shape) and (2) the problem is studied in a non-convex and non-smooth setting. Thus we obtain characterization results for finite unions of Wulff shapes (bubbling) within the class of mean-convex sets or even for general sets with positive reach. Several related results are established. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a uniformly convex normed vector space, formulas for the derivative of the localized volume function of a compact set and general versions of the Heintze--Karcher inequality.

(Based on joint work with Mario Santilli)

(Contact: J. Hörrmann)

## July 14, 2022

*14:00 -16:00 (A&G),*

**Anne Marie Svane** (Assistenzprofessorin aus Aalborg in Dänemark)

**Analyzing point processes using Ripley’s K-function and persistence homology**

*Abstract: *The first half of this talk will give an introduction to two tools for analyzing point patterns, namely Ripley’s K-function and persistence homology. A brief introduction to some of the most common stochastic models for point patterns will also be given. When analyzing the goodness of fit of a given point pattern to a hypothesized stochastic point process model, the K-function and persistence Betti numbers can be used as summary statistics. In the second half of the talk, I will present functional central limit theorems for both the K-function and persistence Betti numbers and show how these results can be used for doing statistics of point processes.

This is joint work with Christophe Biscio, Nicholas Chenavier, and Christian Hirsch.

(Contact: J. Hörrmann)

## July 21, 2022

*14:00 -16:00 (A&G),*

**Chiara Meroni** (Max-Planck Institut Leipzig)

**Intersection bodies using algebra**

*Abstract: *I will discuss the notion of intersection bodies, important constructions in convex geometry. The idea is to approach them using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is a semialgebraic set and provide an algorithm for its computation. This is a joint work with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.

(Contact: J. Hörrmann)

## August 1, 2022

*14:00 -16:00 (A&G),*

**Manuel Quaschner** (Friedrich-Alexander-Universität Erlangen-Nürnberg)

**Non-collision singularities in ***n*-body problems

*n*-body problems

*Abstract: *The existence of non-collision singularities in the $n$-body problem was already conjectured by Painlevé in 1895. Even before the existence was proven in the 1990s, the question came up, whether the set of all initial conditions leading to non-collision singularities is a set of measure 0. A first result of this kind was proven for $n=4$ particles in $d\geq 2$ dimensions by Saari (1977). Using the so called Poincaré surface method, Fleischer (2018) could improve this for $n=4$ particles in $d \geq 2$ dimensions by extending the result to a wider class of potentials. But the problem is still open for more than four particles. \\

After an overview of these works, we consider trajectories that are close to the case of four particles but with some small perturbing forces given by additional particles. In order to apply the Poincaré surface method, we need to prove that these forces cannot break the bounds derived for the four-particle case. If time allows, we will sketch in the end how one could use these results to prove the improbability of larger systems that can be suitably decomposed into diverging systems of up to four particles each.

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(Contact: Prof. Matveev)

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