**Dates:** Every Wednesday, October 9, 2023 - February 2, 2024

**Organizers: **Christoph Thiele and Christian Brennecke

**Venue:** Lipschitzsaal, Mathezentrum, Endenicher Allee 60, 53115 Bonn

**Date**

**Hausdorff Tea**

**Hausdorff Colloquium**

**Graduate Colloquium**

06.12.23

15:00

20.12.23

15:00

15:15

Luise Puhlmann (University of Bonn):

Improved guarantees for the a priori TSP

10.01.24

15:00

### Abstracts

**Elena Demattè (University of Bonn): Entropy: can increasing disorder be useful?**

Starting from its physical meaning, in this talk we will introduce the concept of entropy. Using the classical example of the Boltzmann equation for dilute gases we will see how the entropy suggests the "one-way direction" of time and how this creates apparently a paradox for a system which should be described by deterministic and reversible laws. Based on this example we will also investigate the role of entropy in mathematics as a useful tool in the studies of PDEs.

**László Erdős (IST Austria): New Universality Results for Random Matrices**

Large random matrices tend to exhibit universal fluctuations. Beyond the well-known Wigner-Dyson and Tracy-Widom eigenvalue distributions, we overview other universality results for Hermitian and non-Hermitian matrices. We discuss the emergence of normal distribution involving eigenvectors, especially the random matrix version of quantum unique ergodicity. We also explain why results on non-Hermitian random matrices are much harder than their Hermitian counterparts and highlight our new methods to tackle them.

**Manuel Esser (University of Bonn): How to model evolution? - An individual-based perspective**

The biological theory of adaptive dynamics aims at studying the interplay between ecology and evolution through the modeling of the basic mechanisms: heredity, mutations and competition. A rigorous derivation of the theory was achieved over the last two decades in the context of stochastic individual-based models. These stochastic processes (ess. Markov processes) are driven by microscopic interactions between single individuals and evolve over time towards states of higher fitness. The typical evolutionary behaviour can be studied by looking at limits of large populations and rare mutations. This talk introduces carefully the individual-based model of adaptive dynamics and gives an overview of basic but interesting results as well as the open questions in this field. It is is based on an ongoing collaboration with Anna Kraut and previous works of Anton Bovier, Lorene Coquille and Charline Smadi.

**Jürg Fröhlich (ETH Zürich): After almost a century of Quantum Mechanics, how big is the confusion?**

In 2025, Quantum Mechanics will be 100 years old. Yet, the question as to its deeper meaning seemingly remains unanswered and causes enormous confusion. The goal of my lecture, is to contribute to alleviating this jumble by suggesting an answer. To prepare the ground I will recapitulate the celebrated Kochen-Specker theorem concerning the non-existence of hidden variables in quantum mechanics. I will continue by recalling Gleason’s theorem and Bell’s Inequalities. As the most important part of this lecture I sketch a specific proposal of a completion of Quantum Mechanics. It is based on a precise notions of “events” giving rise to a general principle and a new postulate. My proposal clarifies the role of randomness in Quantum Mechanics, and it leads to a solution of the so-called ”measurement problem.

**Eero Saksman (University of Helsinki): What is multiplicative chaos?**

We give an introduction to basic ideas behind Gaussian multiplicative chaos (GMC). These random measures are obtained as exponentials of random irregular functions, like the Gaussian free field. Also, some applications of GMC will be described.

**Radu Toma (University of Bonn): Integer points on spheres throughout the centuries**

Which numbers are the sum of two integer squares? How about three squares? In fact, can we estimate the number of ways we can represent numbers are sums of k squares? Geometrically, that’s just counting integer points on spheres. Let’s say we can answer the previous questions and it turns out that there are many such points, as the radius grows. Can we say something about the way they distribute on the sphere? Answers to such questions were boasted about by Fermat, had Gauß write EURYKA in his notebook, and involve beautiful and diverse mathematics, inspiring research still to this day. I will give some flavour of the ideas involved, roughly following the history of the problems.

**Mauro Varesco (University of Bonn): An introduction to the Hodge conjecture**

The Hodge conjecture poses a fascinating question about the inherent algebraic properties of geometric shapes. Through notions such as cohomology rings and algebraic cycles, I will give an introduction to the deep connection between algebra, topology, and complex analysis.

**Simone Warzel (TU Munich): Towards a mathematical description of quantum spin glasses**

Classical spin-glass models such as Sherrington-Kirkpatrick's are paradigms for complex disordered systems. In 1980, Parisi described their thermodynamic behaviour using a novel order parameter that captures the emergent hierarchical spin-glass order at low temperatures. The fate of this low-temperature phase with respect to quantum effects induced, e.g., by a transverse magnetic field is a fundamental question. In this talk, I will provide an overview of recent progress in the mathematical description of such quantum glasses. In particular, I will describe what a quantum version of Parisi's famous variational formula looks like.