Algebraic Geometry

The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a…

The main theme of the talk is the combinatorics of lattice polygons and its relationship to the geometry of the associated toric surfaces. Our point of view is to measure the complexity of lattice polygons via the complexity of geometric objects to which they give rise. For the latter, we will…

The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau…

Fano varieties are one of the three building blocks of algebraic varieties.In this talk, we will discuss how to describe a general n-dimensional Fano variety.Although there is no consensus on how to answer to this question, we will explore some new invariants motivated by…

Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory…

I'll start by defining the Chow ring, which is an important invariant of a scheme (or stack). Next, I will define the Picard variety and Picard stack of a curve, and then introduce their universal versions $J^d_g$ and $\mathscr{J}^d_g$ over the moduli space of curves $M_g$. Recently, progress…

I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions.  One can view this as a “higher” version of the Mordell-Weil group for families of elliptic curves, giving rise to sections of “higher” Jacobian…

I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions.  One can view this as a “higher” version of the Mordell-Weil group for families of elliptic curves, giving rise to sections of “higher” Jacobian…

The well-known Adams conjecture in topology is a theorem about compactifications of real vector bundles on CW-complexes, which has important implications for analyzing stable homotopy groups of spheres. In the talk we will discuss an algebro-geometric version of this statement, which tackles…

In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities.…