Research interests
Most of my research is on geometrical problems that arise from the study of surfaces. Below I explain some of my projects.
Most of my research is on geometrical problems that arise from the study of surfaces. Below I explain some of my projects.


Squaretiled surfaces
A squaretiled 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 squaretiled surfaces is to find a good way to characterize them with respect to their geometric properties. The simplest example of a squaretiled 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. Lifting this action, one obtains the action of SL(2,Z) on the set of all squaretiled surfaces. The second question is to classify squaretiled 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 squaretiled surfaces. Surfaces of genus 2 that cover the square torus have at most two branch points and the relative position of these points on the square serves as 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 squaretiled 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, called 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 relating properties of such algebraic curves to translation surfaces that exhibit these symmetries. In other words, a high order automorphism of an algebraic curve has an eigenbasis in the space of holomorphic 1forms on that curve. Every 1form can be used to unfold the algebraic curve into Euclidean plane via integration obtaining a translation surface. My goal is to present eigenbases of high order automorphisms on algebraic curves as translation surfaces and understand the connection between Euclidean metric and algebrogeometric properties of such curves, such as the Jacobian. 