Let G be a connected linear real reductive group with a maximal compact subgroup K. In this talk, we will discuss an approach to studying the large-time behavior of the heat kernel on the corresponding homogeneous space G/K using Bismut's formula. We will try to explain how Bismut's formula provides a natural link between the index theory and representation theory.  In particular, Vogan's λ-map in the representation theory of G plays a central role in the large time asymptotic analysis about the trace of the heat kernel. This talk is based on joint works with Shu Shen and Yanli Song. 

Mean curvature flow (MCF) is the deformation of surfaces with velocity equal to the mean curvature vector. MCF originated in materials science and is widely used as a tool for geometric and topological problems. Major open questions about MCF include how large of singular sets can form, whether the area of the flow is continuous through singular times, and how the various weak solutions may differ. We address these questions under an assumption on the size of the set of singularities with “slow” mean curvature growth. With this assumption, an n-dimensional flow has H^n-measure zero singular sets at every time, has mass that is continuous through singular times, and under an additional mild condition, the level set flow fattens at the discrepancy time of the inner/outer flow. The key technical development is a generalized Brakke equality, which characterizes the deviation from equality in Brakke’s inequality. This is achieved by developing a worldline analysis of Brakke flow, which allows us to relate the regular parts of the flow at different times and estimate the transport of mass into and out of the singular set.