My research is in the field of algebraic geometry, which studies the solutions of polynomial equations. I am interested in special systems of equations which can be characterized geometrically as satisfying a certain positivity condition (the technical condition is that the anticanonical class is big). Conjecturally at least, this condition is equivalent to the arithmetic statement that the corresponding system of equations has "lots" of solutions in rational numbers (technically you have to be willing to pass to a larger number field for this statement to be true). My favorite varieties (the technical term for the space of solutions to a system of polynomial equations) to study are generalized Del Pezzo surfaces, which have a rich history dating back to the 19th century. The most well-known of these surfaces are the cubic surfaces in projective 3-space, given as the zero locus of a single homogeneous cubic polynomial in four variables. The discovery that these surfaces contain 27 lines was one of the landmark achievements of algebraic geometry in the 1800s. These surfaces still hold secrets that have yet to be unlocked.

In many ways, these systems of polynomial equations behaves like projective space. Projective space can be characterized algebraically through its graded homogeneous coordinate ring. The varieties I am in interested, particularly generalized Del Pezzo surfaces, have an analogous multigraded homogeneous coordinate ring that encodes a lot of geometric information about the variety. One of my main projects is trying to understand how to decode all of the information that is contained in this coordinate ring.

It turns out that the multigrading itself gives information about the effective cone of the variety. This is a cone which describes what curves lie on the surface. For generalized Del Pezzo surfaces, this cone turns out to have a beautifully symmetric structure related to a root system associated to the surface. Joint work with Zach Teitler and Ulrich Derenthal has explored some of this symmetry. Ongoing work continues to apply this beautiful symmetry to conjectures about solutions of the corresponding system of polynomial equations in rational numbers.