Manifolds, homotopy theory, and related topics

Hurwitz spaces are important moduli spaces in both number theory and algebraic geometry. From my joint work with Aaron Landesman, I will explain how Hurwitz spaces associated to finite racks satisfy homological stability, as well as modules over these Hurwitz spaces associated to more general surfaces. I will moreover explain how the stable homology of such spaces can be described in terms of that of simpler ones.

Moduli spaces of manifolds are important objects in geometric topology, playing a central role in the classification of families of manifolds. Recently, Galatius and Randal-Williams provided a complete description of the homology of stable moduli spaces of even dimensional manifolds. In this talk, I will present an odd dimensional analogue and explain the key steps in the proof of this result.

I will explain my work on extending Kontsevich's construction of characteristic classes in terms of configuration space integrals to fibre bundles with more general fibres.

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