I am interested in areas of mathematics where algebra and geometry commingle. I am particularly interested in arithmetic and algebraic geometry. My thesis research is on integral models of Shimura varieties. I have also been working in geometric group theory.
A braid is a bijection of a finite set to itself using strands, where we keep track of the crossings and a strand cannot double back. The set of braids on n strands forms a group under composition. The braid group on n strands also arises geometrically as the mapping class group of the disk with n punctures. Roughly, the mapping class group of a manifold M is the group of homeomorphisms from M to M under composition, where we consider two such homeomorphisms the same if one can be deformed into the other. In the 1930s, Burau used this geometric description to construct a representation of the braid group on n strands. For n=2 or 3 the Burau representation is unfaithful, whereas work in the past 3 decades has shown it is unfaithful for n larger than 4. Now n=4 is the last remaining open case.
The faithfulness of the Burau representation on 4 strands can be investigated using a criterion involving certain arc pairs in the 4 punctured disk. Finding an arc pair with certain intersection properties will show that the representation is unfaithful, whereas it is faithful if none exist. Myself and N. Fullarton investigated these arc pairs and were able to show via computer program that no such arc pairs satisfying the requirements exist with at most 2000 intersections. We also studied the set of arc pairs and found remarkable behavior which we exploited to give further numerical evidence that the Burau representation on 4 stands is faithful.
The interaction between elliptic curves and number theory dates back to the latter part of the 19th century. The higher-dimensional analogue of an elliptic curve is an abelian variety, and in the 1960s Shimura generalized the study of moduli spaces of elliptic curves to that of moduli spaces of abelian varieties. These moduli spaces, now known as PEL Shimura varieties, play an integral part in the Langlands program.
My research focuses on the construction of "local models" for certain types of Shimura varieties. A local model is a space which controls singularities of the Shimura variety but is somehow easier to study. Any local property of the Shimura variety can be investigated on the local model. This particular construction gives explicit equations that can be written down.
The local model can be used to obtain resolutions of the Shimura variety. Such a resolution is constructed in my thesis for the Shimura variety associated with GSp_4 and level subgroup given by the pro-unipotent radical of an Iwahori. One first constructs a resolution of the local model and then carries out a similar process on the Shimura variety. The picture to the right is the "dual complex" of the special fiber of the resolution of the local model. It describes how the irreducible components of the special fiber intersect.