I will report on work joint with Florian Naef in which we produce, for a map f of spaces over a space B such that f has compact fibers, a rational model for the fiberwise transfer of fiberwise topological Hochschild homology, considered as a map of parametrized spectra over B. This is motivated by applications to moduli spaces of manifolds: in particular we can detect the vanishing of certain cohomology classes originating from a graph complex via the classifying space of homotopy automorphisms.
 

In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.