Department of Mathematics
San Francisco State University
Abstract: Broadly, my research interests are in algebraic geometry. More specifically, I study moduli spaces of curves, maps, and sheaves, with an emphasis on investigating the mathematical realizations of physical dualities suggested by string theory.
Informal Overview: Roughly speaking, algebraic geometry investigates the geometry of certain objects by studying the algebraic structure of the polynomial functions on those objects. By some measure, the most basic objects in algebraic geometry are the one-dimensional objects, called curves. Confusingly, if you work over the two-dimensional complex numbers (I typically do), then a smooth curve actually looks (topologically) like a surface.
Despite the overly simplified picture above, the complex curve knows so much more than just the underlying topological surface; it also knows all about the complex algebraic functions on the surface. We call this additional data the complex algebraic structure of the curve. Perturbing the algebraic structure on a fixed topological surface leads to the notion of moduli. The moduli spaces of curves are complex algebraic objects that parametrize all such algebraic structures. These moduli spaces are fascinating objects from a purely mathematical perspective.
Starting in the 1990's, physicists studying string theory have also been drawn to the study of complex curves and their moduli. To a physicist, a complex curve represents the history, or worldsheet, of a string propogating through space-time.
By studying physical dualities such as mirror symmetry, where one physical system is described in multiple ways, physicists have made some very compelling conjectures about moduli spaces of curves (and related moduli spaces of maps, sheaves, etc.). These conjectures have shed a whole new light on old questions in algebraic geometry, while raising many new and exciting directions in the field.
Fast forward a few decades: the connections between theoretical physics and pure mathematics have matured greatly. A few of the original conjectures have been solved, but many new ones have been developed, and the majority remain to be understood or verified mathematically. A significant part of my current work, along with a large and growing group of mathematicians and physicists all over the world, focuses on better understanding the pure mathematical implications of the physical dualities that arise in the study of string theory.
Further Details: If you would like to know more about the particular problems I have worked / am working on, have a look at the my papers or contact me directly so we can schedule a time to meet.