# Research

Here are the papers I have written that relate to my current research
interests in mirror symmetry:

Zeta functions of alternate
mirror Calabi-Yau families,
(with Charles F. Doran,
Adriana Salerno,
Steven Sperber,
John Voight,
and Ursula Whitcher).

- Abstract: We prove that if two Calabi-Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard-Fuchs differential equation. The factor in the zeta function is definable over the rationals and has degree equal to the order of the Picard-Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.

Fractional Calabi-Yau Categories from Landau-Ginzburg Models, (with David Favero), to appear in Algebraic Geometry.

- Abstract: We give criteria for the existence of a Serre functor on the derived category of a gauged Landau-Ginzburg model and describe it explicitly. This is used to provide a general theorem on the existence of an admissible (fractional) Calabi-Yau subcategory of a gauged Landau-Ginzburg model. Our framework is completely explicit in the toric setting. As a consequence, we generalize several theorems and examples of Orlov and Kuznetsov, ending with new examples of semi-orthogonal decompositions containing (fractional) Calabi-Yau categories.

Derived Categories of BHK Mirrors, (with David Favero).

- Abstract: We prove a derived analogue to the results of Borisov, Clarke, Kelly, and Shoemaker on the birationality of Berglund-Hubsch-Krawitz mirrors. Heavily bootstrapping off work of Seidel and Sheridan, we obtain Homological Mirror Symmetry for Berglund-Hubsch-Krawitz mirror pencils to hypersurfaces in projective space.

Equivalences of Families of Stacky Toric Calabi-Yau Hypersurfaces, (with Charles F. Doran and David Favero).

- Abstract: Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartics in projective 3-space, a birational reduction of Reid???s list to 81 families, and illustrations of Hodge-theoretic jump loci in toric varieties.
- The code for the computations in this paper can be found here.

Proof of a Conjecture of Batyrev and Nill, (with David Favero), to appear in the American Journal of Mathematics.

- Abstract: We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill. The proof involves passing to an associated category of singularities and toric variation of geometric invariant theory quotients.

Picard Ranks of K3 Surfaces of BHK Type, Fields Institute Monographs, Calabi-Yau Varieties: Arithmetic, Geometry and Physics 34 (2015), 45-63.

- Abstract: We give an explicit formula for the Picard ranks of K3 surfaces that have Berglund-H??bsch-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-H??bsch-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.

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.