Research

I study algebraic geometry, a field that uses equations to understand geometric shapes and the ways they can vary in families, called moduli spaces. One of the central problems in this area is how to enlarge a family of smooth geometric objects by adding meaningful limiting objects that may develop singularities. There are often several different ways to do this, each capturing different aspects of the geometry.

My research aims to understand how these different enlargements are related. I use geometric methods to describe what happens at their boundaries, how geometric objects change as one moves from one enlargement to another, and why constructions coming from seemingly different parts of mathematics can sometimes produce the same space.

My research in algebraic geometry lies in the intersection of birational geometry and moduli theory where we try to classify algebraic varieties up to isomorphism. Given a class of smooth algebraic varieties, one can construct a moduli space that paramterizes it. However, these spaces are challenging to study from a geometric perspective as they are not compact and one can provide a compactification that gives geometrically meaningful singular degenerations of the varieties parameterzed in the interior to the boundary. 

However, there are often many moduli theories offering different compactfications of the same moduli space, such as KSBA stable pairs, K-stability, GIT, and Hodge-theoretic compactifications. Often times, these approaches have a wall crossing theory providing a wall and chamber decomposition in which the compactification stays isomorphic within the same chamber while there are geometric maps between chambers, called wall crossing morphisms, that tells us how the different compactifications are related. 

My work focuses on using explicit methods to describe and compare compactifications of moduli spaces of algebraic surfaces, with a particular focus on KSBA wall crossing and its relationship with other modular compactifications coming from GIT and Hodge theory. 

My research concerns the birational geometry of modular compactifications of algebraic surfaces. I am particularly interested in comparing compactifications arising from KSBA theory, GIT,  K-moduli and Hodge-theoretic constructions, and in understanding how these models are related by wall-crossing and the log minimal model program.

The guiding problem is to understand the network of birational models attached to a given moduli problem. Different stability conditions produce different boundary degenerations and modular interpretations, while arithmetic or Hodge-theoretic constructions can give apparently unrelated compactifications of the same open moduli space. I study when these models coincide, how their boundary stratifications correspond, and which explicit birational transformations—such as blow-ups, flips, or changes of stable replacement—connect them.

I use the explicit geometry of degenerations, boundary divisors, group actions, invariant cones, and period maps to make wall-crossing and compactification theory concrete. A broader goal of my research program is to develop higher-dimensional Hassett–Keel–Looijenga- pictures in which KSBA or K-moduli wall-crossing interpolates between birational, GIT, and arithmetic compactifications. Right now, I am currently applying this research program to the moduli of cubic surfaces with a marked line and the moduli of Wehler K3 surfaces.