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 currently study are related to understanding Hodge theoretic invariants of algebraic varieties that have discrete or combinatorial origins.
"Modularity of Landau-Ginzburg models" (with C. Doran, L. Katzarkov, M. Ovcharenko, and V. Przyjalkowski), to appear, Proc. Steklov Inst. Math. (arXiv)
"Motivic geometry of 2-loop Feynman integrals" (joint with C. Doran and P. Vanhove) to appear, Quart. J. Math. (arXiv)
"Tori and the weight filtration". to appear, Proc. Amer. Math. Soc. (arXiv).
"P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds". (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" (with A. Thompson). Trans. Amer. Math. Soc. (2020) (arXiv, journal).
"Perverse sheaves of categories and some applications" (with L. Katzarkov). Adv. Math. (2019) (arXiv, journal).
"Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces" (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" (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" (with C. Doran). Canad. J. Math. (2016) (arXiv, journal).
"Calabi-Yau threefolds fibred by Kummer surfaces associated to products of elliptic curves " (with C. Doran, A. Novoseltsev, and A. Thompson). In String-Math 2014 proceedings (2016) (arXiv).
"Calabi-Yau threefolds fibred by mirror quartic K3 surfaces" (with C. Doran, A. Novoseltsev, and A. Thompson). Adv. Math. (2016) ( arXiv, journal).
"Families of lattice polarized K3 surfaces with monodromy", (with C. Doran, A. Novoseltsev, and A. Thompson). IMRN (2015) (arXiv, journal).
"Perverse sheaves of categories and non-rationality" (with L. Katzarkov and Y. Liu). In Geometry over Non-Closed Fields (Simons Symposia in Mathematics) (2017), (journal).
"Picard-Fuchs uniformization and modular subvarieties" (with B. Doran and C. Doran). In Uniformization, Riemann–Hilbert Correspondence, Calabi–Yau Manifolds, and Picard–Fuchs Equations (2017).
"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).
"Calabi–Yau threefolds fibered by high rank lattice polarized K3 surfaces", GADEPS, August 2020 (link)
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.