Veranstaltungen am Mathematischen Institut

Abstract: This talk in the area of low-dimensional topology deals with the problem of determining the topological 4-genus for the special case of 3-braid knots. The 4-genus of a knot is the minimal genus of a "nicely" embedded surface in the 4-dimensional ball with boundary the given knot. Asking whether a knot has 4-genus zero, i.e. whether it bounds a disk in the 4-ball, is a natural generalization in dimension 4 of the question whether it is isotopic to the trivial knot. It is one of the curiosities of low-dimensional topology that constructions such as finding these disks can sometimes be done in the topological category, but fail to work smoothly. The first examples of this phenomenon followed Freedman's famous work on 4-manifolds. Four decades later, the topological 4-genus of knots - even torus knots - remains difficult to determine. In a joint work with S. Baader, L. Lewark and F. Misev, we classify 3-braid knots whose topological 4-genus is maximal (i.e. equal to their 3-genus). In the talk we will explain the difficulties that arise and draw connections to other problems in low-dimensional topology. We will define all relevant notions and address a broad audience.

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 $G$.

Abstract: What has homological mirror symmetry ever done for you? I will give my personal answer to that question and discuss joint work in progress with Liam Watson and Artem Kotelskiy concerning the behaviour of Khovanov homology under Conway mutation.