Veranstaltungen am Mathematischen Institut

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 present a graph-theoretic model for dynamical systems given by a homeomorphism f on the Cantor set X. This construction gives a bijective correspondence between such dynamical systems (X,f) and a subclass of two-colored Bratteli separated graphs. We use this construction in order to write any dynamical system of our interest as an inverse limit of a sequence of (what we call) generalized finite shifts. This enables us to compute the associated Steinberg algebra (resp. C*-algebra) of the dynamical systems as colimits of the graph algebras (resp. graph C*-algebras) associated with the different levels of the corresponding separated graph. In subsequent work we plan to apply this theory to relate the type semigroup of the dynamical system with the graph monoid of the corresponding separated graph, and with the non-stable K-theory of the Steinberg algebra. This is joint work with Pere Ara (Universitat Autònoma de Barcelona).

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.