Oberseminare und sonstige Vorträge

In this talk we dive into the world of point-line geometries related to spherical buildings of type F4. These turn out to be parapolar spaces of rank 3 with some extra properties. They differ from the other exceptional spherical buildings by the fact that they are not determined by only a field, but one also needs a quadratic alternative division algebra over this field. This makes them a bit harder to tackle and therefore they are omitted by several authors. After a short introduction, we will discuss some analogs of recent results about other exceptional spherical buildings. In particular, we will discuss subgeometries, domestic collineations and kangaroos

Extreme black holes possess event horizons at zero temperature, referred to as degenerate Killing horizons. These horizons are exclusively delineated by a specific limiting procedure, defining a near-horizon geometry or, more broadly, a quasi-Einstein equation which governs their properties. Solutions to this equation manifest as triples (M, g, X), where M represents a closed manifold (the horizon), g denotes a Riemannian metric, and X is a 1-form. The talk will be a overview of these concepts and relevant results which characterize solutions to the quasi-Einstein equation. This is joint work with Eric Bahuaud, Hari Kunduri, and Eric Woolgar.

Directed graph algebras have long been studied as tractable examples of C*-algebras, but they are limited by their inability to have torsion in their K_1 group. Graphs of groups, which are famed in geometric group theory because of their intimate connection with group actions on trees, are a more recent addition to the C*-algebra scene. In this talk, I will introduce the child of these two concepts directed graphs of groups and describe how their algebras inherit the best properties of its parents, with a view to outlining how we can use these algebras to model a class of C*-algebras (stable UCT Kirchberg algebras) which is classified completely by K-theory.

Abstract: Synthetic homotopy theory is a general framework for constructing interesting contexts for doing homotopy theory: using the data of a spectral sequence in some category $\mathcal{C}$, one can construct another category which can be viewed as a deformation of $\mathcal{C}$. The motivating example is the fact, due to Gheorghe-Wang-Xu, that ($p$-complete, cellular) $\mathbb{C}$-motivic homotopy theory can be described as a deformation of the ordinary stable homotopy category, simply using the data of the Adams-Novikov spectral sequence. Burklund, Hahn, and Senger used this framework to study $\mathbb{R}$-motivic homotopy theory as a deformation of $C_2$-equivariant homotopy theory. In joint work with Gabe Angelini-Knoll, Mark Behrens, and Hana Jia Kong, we give (up to completion) a different synthetic description of this deformation, which generalizes to give a deformation of (Borel-complete) $G$-equivariant homotopy theory for other groups $G$.

We talk about the Classification of Finite Simple Groups from (at least) two perspectives: Where does it come from? How is it applied? What is special about this big result, both in terms of its origin and proof and in terms of what we can learn about how mathematical work changed during that time?