Sid Maibach (University of Bonn): An Exposition to Random Conformal Geometry

In this talk I will present an overview of the research area sometimes titled "random conformal geometry". This is the study of random objects that are symmetric under holomorphic coordinate changes, making them well-defined objects on Riemann surfaces. The key objects I will introduce are "Schramm–Loewner evolution" random curves, "Gaussian free field" random functions, and concepts from conformal field theory. These appear universally in the study of systems at critical temperature on 2D lattices as the spacing of the lattice goes to zero. As an example of such a system I will discuss the Ising model. I will also briefly touch upon my own research questions about the correspondence between Kähler structures on the moduli spaces of the underlying Riemann surfaces and large deviation principles for the aforementioned random objects. However, instead of going into detail I will illustrate more examples of probabilistic constructions from which these universal objects emerge.

Antonia Ellerbrook (University of Bonn): Cost Allocation for Set Covering: the Happy Nucleolus

Imagine you were a delivery service operator who wants to visit a certain set of customers. There is a given set of possible tours. Each tour serves a subset of the customers and has a certain cost. You can use as many of these tours as you like. Your task is to set a price for each customer. Of course, you want to charge as much money as possible, but without losing customers. We assume that any group could leave your delivery service and self-fund one of the given tours. Thus, the summed prices of customers in this group should not exceed the cheapest cost for a tour of their own. From here, we will build on previous work in the field of cooperative game theory and develop a fair cost allocation concept with efficient computation.

Sil Linskens (University of Bonn):  Grothendieck's Homotopy Hypothesis

In 1983, Grothendieck wrote the influential manuscript Pursuing stacks. In this work he formulated the famous homotopy hypothesis: “homotopy types = infinity-groupoids”. This was a deep insight, which completely changed our understanding of the place of homotopy theory in broader mathematics. In this talk I will motivate and contextualise the homotopy hypothesis, and then explain its development since 1983.

Thomas Nikolaus (University of Münster): Higher Algebra and Lifts to the Sphere

This talk is about higher algebra, i.e. about ring spectra.
Ring spectra are a higher categorical version of the notion of commutative rings and play an important role in in modern stable homotopy theory. We hope to be able to transport some of the relevant ideas, motivations and applications.The initial object in this world, i.e. the higher cousing of the ring of integers, is the sphere spectrum. It is an old question in homotopy theory which ordinary rings admit a lift to the sphere spectrum. We will explain some new approaches to studying this question and how to understand maps between such lifts. In particular we will see that `number theory over the sphere' looks very different to ordinary number theory.

Alexander West (University of Bonn): Minimizing the Willmore energy under a total mean curvature constraint 

The Willmore energy of a closed surface is the integral of the square of the mean curvature. It appears for example as the main term in the Helfrich energy, used to describe the bending energy of lipid bilayer cell membranes. Consequently, the minimization of the Willmore energy under various constraints has been studied extensively in the past few decades. In this talk, we consider the minimization of the Willmore energy in the class of surfaces with prescribed genus, while keeping a constraint on the total mean curvature and the area of the surface. This problem admits smooth minimizers for an arbitrary genus and a large class of constraints and we will talk about how this existence result can be obtained.

Robert Seiringer (Institute of Science andTechnology Austria): Spectral analysis of polaron models in the strong coupling limit

The Fröhlich polaron and related models of quantum field theory have played a prominent role in mathematical physics over several decades. In this talk, we shall explain recent bounds on the quantum corrections to the (classical) Pekar approximation of the ground state energy of the Fröhlich polaron model in the strong coupling limit, and their consequence on the existence of excited states and the polaron's effective mass.

Hendrik Baers (University of Bonn): Instability of the Fractional Calderón Problem

The Calderón problem is one of the classic examples of an inverse problem. It is about determining the conductivity of a medium by making voltage and current measurements on its boundary. We consider the fractional formulation of the problem and prove exponential instability. Physically that means, small differences in the measurements can lead to very different conclusions about the conductiviy. We will start by introducing the classical formulation of the Calderón problem and then, motivated by this, formulate the fractional version. We will compare some known results and then, finally, we will proceed to the proof of the instability result. There we will see two rather very different mechanisms leading to instability.

Ilya Chevyrev (University of Edinburgh): Stochastic quantisation of gauge theories

In this talk I will present stochastic quantisation (SQ) as a way to rigorously study Euclidean quantum field theories. I will in particular discuss recent progress in the analysis of the stochastic quantisation equations of 2- and 3-dimensional gauge theories. It turns out that these equations, which are systems of singular stochastic partial differential equations, can be renormalised to admit local-in-time solutions that respect the underlying symmetries of the theory. Furthermore, in the case of “pure" Yang-Mills theories in dimension 2, one can show that the associated Yang-Mills (Gibbs) measure is invariant for the dynamic. Although this 2D Yang-Mills measure is exactly solvable, the study of the dynamic can surprisingly reveal new properties of the measure that were not known by other methods. Based on joint works with Ajay Chandra, Martin Hairer, and Hao Shen.

Jeremy Avigad (Carnegie Mellon University): A formal perspective on mathematical structures

Reasoning about axiomatically characterized abstract structures has been central to mathematics since the early twentieth century. Mathematicians today are using the Lean interactive proof assistant to build a formal library called Mathlib, and the ability of the system to manage a complex network of such structures has been essential to its success. In this talk, I will discuss some of the challenges that structural reasoning brings and how they are addressed in Lean and Mathlib.