My research is in combinatorial and computational algebraic geometry. I am particularly interested in toric varieties, their applications to sparse polynomial systems, combinatorics and geometry of polyhedra, and coding theory. Recently, I have been also working on geometric inequalities involving mixed volumes.



   

  • Papers and Preprints

Plücker-type inequalities for mixed areas and intersection numbers of curve arrangements, (with Gennadiy Averkov), preprint (2021), [arXiv]

𝔽_q-zeros of sparse trivariate polynomials and toric 3-fold codes, (with Kyle Meyer and Jenya Soprunova), preprint (2021), [arXiv], see also GitHub for code

Coding theory package for Macaulay2, (with Taylor Ball, Eduardo Camps-Moreno, Henry Chimal-Dzul, Delio Jaramillo-Velez, Hiram López, Nathan Nichols, Matthew Perkins, German Vera-Martínez, Gwyneth Whieldon), (2021), Journal of Software for Algebra and Geometry [arXiv]

The dual of an evaluation code, (with Hiram H. López and Rafael H. Villarreal), (2021), Designs, Codes and Cryptography, doi.org/10.1007/s10623-021-00872-w [arXiv]

Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum, (with Gennadiy Averkov and Christopher Borger), Mathematika 66 (2020), no. 4, pp 1003--1027 [arXiv]

Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes, (with Hiram H. López and Gretchen L. Matthews), Designs, Codes and Cryptography, 88 (2020), no. 8, pp 1673--1685 [Springer link]

Classification of triples of lattice polytopes with a given mixed volume, (with Gennadiy Averkov and Christopher Borger), Discrete and Computational Geometry (2020) 191 [open access], see also GitHub for data and code

Wulff shapes and a characterization of simplices via a Bezout type inequality, (with Christos Saroglou and Artem Zvavitch), Advances in Mathematics 357 (2019) 106789 [arXiv]

Criteria for strict monotonicity of the mixed volume of convex polytopes, (with Frédéric Bihan), Advances in Geometry, Vol 19, Issue 4 527-540 (2019) [arXiv]

Generalized multiplicities of edge ideals, (with Ali Alilooee and Javid Validashti), Journal of Algebraic Combinatorics (2018) Volume 47, Issue 3, pp 441--472 DOI: 10.1007/s10801-017-0781-3 [arXiv]

Characterization of simplices via the Bezout Inequality for Mixed volumes, (with Christos Saroglou and Artem Zvavitch) Proceedings of AMS 144 (2016), no. 12, 5333--5340 [arXiv]

Bezout Inequality for Mixed volumes, (with Artem Zvavitch) [arXiv], International Math Research Notices (2016), 2016 (23): 7230--7252

Eventual quasi-linearity of the Minkowski length, (with Jenya Soprunova) European Journal of Combinatorics 58 (2016), pp 107-117 [arXiv]

Multigraded Hilbert function and toric complete intersection codes, (with Mesut Şahin) Journal of Algebra 459 (2016) pp 446-467 [arXiv]

Lattice polytopes in coding theory, Journal of Algebra Combinatorics Discrete Structures and Applications (2015), Vol 2, No 2, pp 85-94 [open access]

Tropical determinant on transportation polytope, (with Sailaja Gajula and Jenya Soprunova) Linear Algebra and its Applications, Vol 475, No 15 (2015) pp 28-44 [arXiv]

On dual toric complete intersection codes, (with Pinar Celebi Demirarslan) Finite Fields and Their Applications, Vol 33 (2015) pp 118-136 [arXiv]

Toric complete intersection codes, Journal of Symbolic Computation, 50, (2013), pp 374-385 [arXiv]

Bringing toric codes to the next dimension, (with J. Soprunova) SIAM J. Discrete Math. 24, no. 2, pp 655-665 (2010) [arXiv]

Toric surface codes and Minkowski length of polygons, (with J. Soprunova) SIAM J. Discrete Math. 23, no. 1, pp 384-400 (2009) [arXiv]

Let me tell you my favorite lattice point problem..., (with M. Beck, B. Nill, B. Reznick, C. Savage, Z. Xu) "Integer points in polyhedra", Contemp. Math. 452, AMS (2008) [pdf]

Global residues for sparse polynomial systems, Journal of Pure and Applied Algebra 209 (2007), no. 2, pp 383-392. [arXiv]

Combinatorial construction of toric residues, (with Amit Khetan) Ann. Inst. Fourier (Grenoble) 55, no. 2 (2005), pp 511-548 [arXiv] (Also check out a combinatorial problem.)

Toric residue and combinatorial degree, Trans. Amer. Math. Soc., 357 (2005), no. 5, pp 1963-1975 [arXiv]

Residues and tame symbols on toroidal varieties, Compositio Math. 140 (2004) no. 6, pp 1593-1613 [arXiv]

On combinatorial coefficients and the Gelfond-Khovanskii residue formula, "Topics in Algebraic Geometry and Geometric Modeling", Contemp. Math., 334, AMS (2003), pp 343-349 [arXiv]

Symbols and residues on surfaces, (2001) This paper is a 2-dimensional version of the Compositio paper, but considers arbitrary surfaces [pdf]

A short proof of the Prime Number Theorem for arithmetic progressions, (1998) This paper was written in response to Don Zagier's paper "Newman's Short Proof of the Prime Number Theorem" Amer. Math. Monthly 104 (1997), no. 8, pp 705-708 [pdf]

Patching theorem in quadratic Serre's conjecture, Fundam. Prikl. Mat. 3 (1997), no. 1, pp 255-261 (in Russian) [abstract]

   

  • Other

Algebraic Curves and Codes, Lecture Notes (2013), in preparation [pdf]

Parshin's symbols and residues, and Newton polyhedra, Ph.D. Thesis (2002), advisor Askold Khovanskii, [pdf]

Computing the global residue in dimension 2, Maple code

Computing the toric residue for four 3-simplices, Maple code (with A. Khetan).

Geometry SOAR 2001, Lecture notes for the SOAR 2001 math summer camp (with Kiumars Kaveh and Jenya Soprunova), [pdf]

My research is partially supported by NSA Young Investigator Award 2010-2011, 2013-2014