Algebraic geometry, especially duli of curves, logarithmic geometry, and tropical geometry

Research Interests

My research area is algebraic geometry - the study of shapes defined using only addition, subtraction, multiplication, and (when we're feeling fancy) division. The shapes that result include lines and planes, circles and spheres, parabolas and paraboloids, as well as the more mysterious things that arise when you make unusual choices for how to add, subtract, and multiply.

More narrowly, I am interested in moduli spaces of curves, that is, spaces whose points represent algebraic curves. It turns out that there are many ways to build such spaces, coming from different choices of curves to include. An intriguing tool for the study of these moduli spaces is the budding field of tropical geometry, which studies the relatively simple piecewise linear shapes built from the operations of addition, subtraction, and minimum. Logarithmic geometry is a bridge between algebraic and tropical geometry that simultaneously keeps track of both algebraic and tropical information. In my thesis, I show that certain log geometric information can be used to construct new moduli spaces of curves together with a canonical "resolution of the rational map" from the Deligne-Mumford moduli stack of curves.