We study Quot schemes of rank 0 quotients on smooth projective curves. These Quot schemes exhibit a rich and highly structured geometry, with formal parallels to the Hilbert schemes of points on surfaces.

In this talk, we first note formulas for the twisted \chi_y-genera with values in tautological line bundles pulled back from the symmetric product via the Quot-to-Chow morphism, and for the associated twisted Hodge numbers. Going further, we give formulas for the level 2 (twisted) elliptic genus for quotients of a vector bundle of even rank. We also discuss the case of level \ell elliptic genera, for higher values of \ell. 

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. 

The Torelli subgroup of the mapping class group of a surface is the subgroup acting trivially on the first homology of the surface.  We will discuss recent joint work with Andrew Putman where we compute the second rational homology of the Torelli group for closed, orientable surfaces of genus at least 6.  Along the way we compute the first twisted homology of the Torelli group with coefficients in the abelianization of the maximal abelian cover of the closed surface.  We also prove some new purely representation theoretic results about certain infinitely presented representations of the symplectic group

Special Lagrangian submanifolds, introduced by Harvey and Lawson, are an important class of minimal submanifolds in Calabi-Yau manifolds. In this talk, I will explain common constructions of special Lagrangians and then a gluing construction of a special Lagrangian in Calabi-Yau manifolds with K3-fibrations when the K3-fibres are collapsing. Furthermore, these special Lagrangians converge to an interval or loop of the base of the fibration at the collapsing limit. This phenomenon is similar to holomorphic curves collapsing to tropical curves in special Lagrangian fibrations and is only a tip of the iceberg of the Donaldson-Scaduto conjecture. This is a joint work with Shih-Kai Chiu.