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C*-algebras provide non commutative analogues of locally compact Hausdorff spaces. In this talk I’ll provide a survey of the large scale project to classify simple amenable C*-algebras, indicating the role played by non commutative versions of topological ideas. No prior knowledge of C*-algebras will be assumed.

The d-dimensional bordism category Cob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. For d=1 and d=2 this category is well understood because we have a complete list of all 1 or 2-manifolds with boundary. In this talk I will argue that the categories Cob_1 and Cob_2 nevertheless carry a lot of interesting structure. 

I will show that the classifying spaces B(Cob_1) and B(Cob_2) contain interesting moduli spaces coming from the combinatorics of how 1 or 2 manifolds can be glued along their boundary. In particular, I will introduce the notion of a "factorisation category" and explain how it relates to Connes' cyclic category for d=1 and to the moduli space of tropical curves for d=2. If time permits, I will sketch how this relates to the curve complex and moduli spaces of complex curves.

Let S be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of the orbit equivalence relation coming from the action of the mapping class group of S on the Gromov boundary of the arc graph of S. This is joint work with Marcin Sabok.

The geodesic flow on hyperbolic finite-volume hyperbolic manifolds is a particularly well-studied dynamical system; this is in part due to its connection to other important dynamical systems on the manifold, as well as orbital counting and other number-theoretic problems related to discrete subgroups of orthogonal groups. In recent years, there has been some interest in generalizing many of the properties of the geodesic flow on finite-volume manifolds to the infinite-volume setting. I will discuss joint work with Hee Oh in which we establish exponential mixing of the geodesic flow on infinite-volume geometrically finite hyperbolic manifolds with large enough critical exponent. Patterson-Sullivan densities and Burger-Roblin measures, the Lax-Phillips spectral gap for the Laplace operator on infinite volume geometrically finite hyperbolic manifolds, and complementary series representations are all involved in both the statement and proof of our result, and I will try to explain how these different objects are related in this setting.

I will discuss recent progress on understanding the tmf-based Adams spectral sequence, where tmf = topological modular forms.  The idea is to generalize the work of Mahowald and others in the context of bo-resolutions.  The work I will discuss is joint with Prasit Bhattacharya, Dominic Culver, and J.D. Quigley.