My area of research is algebraic geometry, the application of geometric methods to understand solutions of systems of polynomial equations. In linear algebra, one learns algorithms for explicitly describing the solutions of a system of linear polynomial equations. When considering nonlinear equations however, such explicit descriptions are usually impossible to obtain. But we can still gain important insights by investigating geometric properties of the solution set. For instance, is it smooth or singular? How many connected components does it have? Perhaps surprisingly, these simple questions lead to a rich and sophisticated theory, with deep connections to other areas of mathematics, including commutative algebra, topology, and complex analysis.
Within algebraic geometry, I am especially interested in curves, singularities, and moduli theory. More specifically, my research aims to establish a systematic construction and classification of geometric compactifications of the moduli space of algebraic curves, and to develop the technical tools needed to study the geometry of these compactifications. While I view this work as fundamental in its own right, it is also important on account of its connection with theoretical physics. A major impetus for recent work in algebraic geometry has been the construction of mathematically rigorous models for ideas from string theory, and compactifications of the moduli space of curves arise naturally in this context.