About My Research

I am an algebraic geometer whose research involves a combination of homological algebra, category theory, commutative and noncommutative algebra, and symplectic geometry. My primary focus is on derived categories of coherent sheaves on algebraic varieties. The essential direction of my research is to study algebra and geometry by way of gauged Landau-Ginzburg models. Traditionally, algebra and geometry are studied together by thinking of solutions to algebraic equations as shapes. Gauged Landau-Ginzburg models are triples consisting of a geometric shape, an algebraic equation, and a choice of symmetries of this data. The name comes from theoretical physics where Landau-Ginzburg models are used to describe tiny strings propagating through space and time.

Publications.

  • Variation of Geometric Invariant Theory Quotients and Derived Categories, (with M. Ballard and L. Katzarkov)
    Journal für die reine und angewandte Mathematik (Crelles Journal) V. 746, pgs 235-303, 2019

  • Equivalences of Families of Stacky Toric Calabi-Yau Hypersurfaces, (with C. Doran and T. Kelly)
    Proceedings of the American Mathematical Society V. 146 N. 11 pgs 4633-4647, 2018

  • Derived Categories of Degree d Hypersurfaces, (with M. Ballard, D. Deliu, U. Isik, and L. Katzarkov)
    Mathematische Annalen V. 371, I. 1-2, pgs 337-370, 2018.

  • A Proof of a Conjecture of Batyrev and Nill, (with T. Kelly)
    American Journal of Mathematics 139.6, pgs 1493-1520, 2017.

  • Fractional Calabi-Yau Categories from Landau-Ginzburg Models, (with T. Kelly)
    Algebraic Geometry V. 5 I. 5, pgs 596-649, 2018.

  • Homological Projective Duality via Variation of Geometric Invariant Theory Quotients, (with M. Ballard, D. Deliu, U. Isik, and L. Katzarkov)
    Journal of the European Mathematical Society 19.4, pgs 1127-1158, 2017.

  • Resolutions in Factorization Categories, (with M. Ballard, D. Deliu, U. Isik, and L. Katzarkov,
    Advances in Mathematics, 295, pgs 195-249, 2016.

  • The Mori Program and Non-Fano Toric Homological Mirror Symmetry, (with M. Ballard, C. Diemer, L. Katzarkov, and G. Kerr)
    Trans. Amer. Math. Soc. 36, pgs 8933-8974, 2015.

  • Griffths Groups for Derived Categories with applications to Fano-Calabi-Yaus, (with A. Iliev and L. Katzarkov)
    Pure and Applied Mathematics Quarterly, V. 10 N. 1, pgs 1-55, 2014.

  • A Category of Kernels for Graded Matrix Factorizations II: Further Implications, (with M. Ballard and L. Katzarkov)
    Journal de Mathematiques pures et appliquees, 2014. V. 102 I. 4, pgs 702-757, 2014.

  • An Orbit Construction of Orlov Spectra, Phantoms, and Knorrer Periodicity, (with F. Haiden and L. Katzarkov)
    Lecture Notes of the Unione Matematica Italiana, Vol. 15, pgs 33-42, 2014.

  • A Category of Kernels for Graded Matrix Factorizations and Its Implications for Hodge Theory, (with M. Ballard and L. Katzarkov)
    Publications Mathematiques de l'IHES, V. 120 I. 1, pgs 1-111, 2014.

  • Orlov Spectra: Gaps and Bounds, (with M. Ballard and L. Katzarkov)
    Inventiones Mathematicae, V. 189 I. 2, pages 359-430, 2012.

  • Hochschild Dimension of Tilting Objects, (with M. Ballard)
    International Mathematical Research Notices V. 11, pgs 2607-2645, 2012.

  • Reconstruction and Finiteness Results for Fourier-Mukai Partners,
    Advances in Mathematics, V. 229, I., pgs 1955-1971, 2012.

  • Characters and q-series in the rational numbers adjoin the square root of 2, (with D. Corson, K. Liesinger, and S. Zubairy)
    Journal of Number Theory, 107, pgs 392-405, 2004.


  • Preprints.

  • Fundamental Factorization of a GLSM, Part I: Construction, (with I. Ciocan-Fontanine, J. Guere, B. Kim, and M. Shoemaker, 87 pages)

  • Kernels for Grassmann Flops, (with M. Ballard, N. Chidambaram, P. McFaddin, and R. Vandermolen, 27 pages)

  • Kernels from Compactifications, (with M. Ballard and C. Diemer, 59 pages)

  • Derived Categories of BHK Mirrors, (with T. Kelly, 27 pages)