Research interests

Most of my research is on geometrical problems that arise from the study of surfaces. My research interests are best described by this diagram. Below I explain some of my projects.

Most of my research is on geometrical problems that arise from the study of surfaces. My research interests are best described by this diagram. Below I explain some of my projects.

Square-tiled surfaces
A square-tiled surface is a collection of unit squares in the plane whose pairs of parallel sides are identified by translations. One can think about them as geometric tilings on topological surfaces. One question about square-tiled surfaces is to find a good way to characterize them with respect to their geometric properties.The simplest example of a square-tiled surface is a single unit square with opposite sides identified. Topologically it is a torus, geometrically it has a flat (or Euclidean) metric with an extra symmetry given by rotation by 90 degrees, algebraically it is an elliptic curve with j=1728. It also can be thought of as a quotient of the complex plane by an integer lattice. The group SL(2,Z) acts on the complex plane and preserves the lattice, therefore it preserves the square torus. The action of SL(2,Z) extends to all square-tiled surfaces. The second question is to classify square-tiled surfaces up to this action. In my thesis I am giving a partial classifications for surfaces of genus 2. |

Spaces of branched covers
I study the geometry of the parameter spaces of branched covers of a torus, or absolute period leaves. This is closely related to square-tiled surfaces. Surfaces of genus 2 that cover the square torus have at most two branching points and the relative position of these points on the square is a coordinate on the absolute period leaf. It's not hard to show that in this case the absolute period leaf is itself a surface that acquires a square-tiled structure. Such absolute period leaves exist for any degree of the cover. In my thesis I described the geometry of the absolute period leaves for prime degrees and showed that interesting structures, which I call pagodas, arise in this case. |

Surfaces with many symmetries
Some hyperbolic surfaces (alternatively, algebraic curves) have exceptionally many automorphisms, or symmetries: they don't come in continuous families of surfaces with the same group of automorphisms. I am interested in finding translation surfaces (or Euclidean pictures of these surfaces) that exhibit these symmetries. In other words, a high order automorphism of an algebraic curve has an eigenbasis in the space of holomorphic 1-forms on that curve. Every 1-form can be used to unfold the algebraic curve into Euclidean plane via integration. My goal is to present eigenbasis of high order automorphisms on an algebraic curve in the form of Euclidean pictures. |