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.

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.