Research
Overview
I study problems in algebraic geometry with an emphasis on those which have origins in theoretical physics and combinatorics or polyhedral geometry.
Most of the problems that I'm interested in these days are related to understanding Hodge theoretic invariants of algebraic varieties that have discrete or combinatorial origins.
Preprints
Published research
"Tori and the weight filtration". to appear, Proc. Amer. Math. Soc. (arxiv).
"P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds". (joint with Z. Li, J. Shen, and Q. Yin) Forum of Math. Sigma (arxiv, journal).
"Hodge numbers of Landau-Ginzburg models". Adv. Math. (arXiv, journal).
"P=W phenomena" (joint with L. Katzarkov and V. Przyjalkowski). Math. Notes (arXiv, journal).
"Pseudolattices, del Pezzo surfaces, and Lefschetz fibrations" (joint with A. Thompson). Trans. Amer. Math. Soc. (2020) (arXiv, journal).
"Perverse sheaves of categories and some applications" (joint with L. Katzarkov). Adv. Math. (2019) (arXiv, journal).
"Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces" (joint with C. Doran, A. Novoseltsev, and A. Thompson). Math. Z. (2019) (arXiv, journal).
"Hodge numbers from Picard-Fuchs equations" (joint with C. Doran and A. Thompson). SIGMA (2017) (journal).
"Mirror Symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds" (joint with C. Doran and A. Thompson). In String-Math 2015 proceedings (2017) (arXiv).
"Toric degenerations and the Laurent polynomials related to Givental's Landau-Ginzburg models" (joint with C. Doran). Canad. J. Math. (2016) (arXiv, journal).
"Calabi-Yau threefolds fibred by Kummer surfaces associated to products of elliptic curves " (joint with C. Doran, A. Novoseltsev, and A. Thompson). In String-Math 2014 proceedings (2016) (arXiv).
"Calabi-Yau threefolds fibred by mirror quartic K3 surfaces" (joint with C. Doran, A. Novoseltsev, and A. Thompson). Adv. Math. (2016) ( arXiv, journal).
"Families of lattice polarized K3 surfaces with monodromy", (joint with C. Doran, A. Novoseltsev, and A. Thompson). IMRN (2015) (arXiv, journal).
"Perverse sheaves of categories and non-rationality" (joint with L. Katzarkov and Y. Liu). In Geometry over Non-Closed Fields (Simons Symposia in Mathematics) (2017), (journal).
"Picard-Fuchs uniformization and modular subvarieties" (joint with B. Doran and C. Doran). In Uniformization, Riemann–Hilbert Correspondence, Calabi–Yau Manifolds, and Picard–Fuchs Equations (2017).
Expository articles
"Introduction to Homological Mirror Symmetry", in SDCD 2016: Superschool on Derived Categories and D-branes (Springer Proceedings in Mathematics & Statistics) (journal).
"The geometry and moduli of K3 surfaces", in Calabi-Yau Varieties: Arithmetic, Geometry and Physics (Fields Institute Monographs) (arXiv, journal).
Unpublished notes
Slides etc.
"Calabi–Yau threefolds fibered by high rank lattice polarized K3 surfaces", GADEPS, August 2020 (link)
Student supervision
Students interested in research projects with me are encouraged to get in touch.
Undergraduates should have, at minimum, background in linear and abstract algebra (Lehigh's 242 and 243), and should have completed Calculus III (Lehigh's Math 023).
Graduate students interested in pursuing a Ph.D. with me should have solid background in algebra, geometry, and topology. At minimum, a graduate student should have; a very solid grasp of groups, rings, and fields; a basic understanding of smooth manifolds or Riemann surfaces; a working knowledge of fundamental groups, homology, and cohomology of smooth manifolds.