Research

Papers I have written that relate to my current research interests in mirror symmetry:

Picard Ranks of K3 Surfaces of BHK Type, to appear in the Fields Institute Monograph Series.

Abstract: We give an explicit formula for the Picard ranks of K3 surfaces that have Berglund-Hübsch-Krawitz (BHK) Mirrors over an algebraically closed field. These K3 surfaces are those that are certain orbifold quotients of weighted Delsarte surfaces. The proof is an updated classical approach of Shioda using rational maps to relate the transcendental lattice of a Fermat hypersurface of higher degree to that of the K3 surfaces in question. The end result shows that the Picard ranks of a K3 surface of BHK-type and its BHK mirror are intrinsically intertwined. We end with an example of BHK mirror surfaces that, over certain fields, are supersingular.

BHK Mirrors via Shioda Maps, Advances in Theoretical and Mathematical Physics, 17 no. 6 (2013), 1425-1449.

Abstract: In this paper, we give an elementary approach to proving the birationality of multiple Berglund-Hübsch-Krawitz (BHK) mirrors by using Shioda maps. We do this by creating a birational picture of the BHK correspondence in general. Although a similar result has been obtained in recent months by Shoemaker, our proof is new in that it sidesteps using toric geometry and drops an unnecessary hypothesis. We give an explicit quotient of a Fermat variety to which the mirrors are birational.
A video of me presenting a talk at String-Math 2013 on this paper can be found here.
The slides for that talk can be found here.

Mirror Quintics, discrete symmetries and Shioda Maps (with Gilberto Bini and Bert van Geemen) Journal of Algebraic Geometry, 21 (2012), 401-412.

Abstract: In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard Fuchs equation associated to the holomorphic 3-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one parameter families to the family of Mirror Quintics. Our constructions generalize to degree n Calabi Yau varieties in (n-1)-dimensional projective space.

Undergraduate Research:

Quiver Grassmannians and their Quotients by Torus Actions, Master's Thesis under Elham Izadi.

On Kostant's Theorem for Lie Algebra Cohomology (with UGA VIGRE Algebra Group), Contemp. Math., 478 (2008), 39-60

Two papers at the Department of Defense on probability and electrical engineering.