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,
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.
Categories from Landau-Ginzburg Models,
(with David Favero), to appear in
- 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)
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.
of Families of Stacky Toric Calabi-Yau Hypersurfaces,
(with Charles F. Doran and
- 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
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.
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
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.