My research area is in real algebraic geometry and its connections with several other fields, especially combinatorics and convex optimization. I'm generally interested in questions involving polynomials, determinants, and matroids. See the publications and talk slides below. Here is a research statement from 2019.

Publications

Publication List, also on arXiv, Google Scholar and MathSciNet.

• Determinantal representations and the image of the principal minor map (with Abeer Al Ahmadieh).
• Moments, Sums of Squares, and Tropicalization (with Greg Blekherman, Felipe Rincón, Josephine Yu)
• Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials (with Abeer Al Ahmadieh).
• Invariant hyperbolic curves: determinantal representations and applications to the numerical range (with Faye Pasley Simon).
• Sparse moments of univariate step functions and allele frequency spectra (with Zvi Rosen and Georgy Scholten).
• Log-Concave Polynomials IV: Approximate Exchange, Tight Mixing Times, and Near-Optimal Sampling of Forests (with Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Thuy Duong Vuong), STOC 2021.
• Positively Hyperbolic Varieties, Tropicalization, and Positroids (with Felipe Rincón and Josephine Yu), Advances in Mathematics, 383 (2021).
• Differential signatures of algebraic curves (with Irina Kogan and Michael Ruddy), SIAM Journal on Applied Algebra and Geometry. 4 (2020), no. 1, pp. 185 -- 226.
• Log-Concave Polynomials III: Mason's ultra-log-concavity conjecture for independent sets of matroids (with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan).
• Log-Concave Polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid (with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan), STOC 2019.
• Log-Concave Polynomials I: entropy and a deterministic approximation algorithm for counting bases of matroids (with Nima Anari and Shayan Oveis Gharan), FOCS 2018.
• Generalized eigenvalue methods for Gaussian quadrature rules (with Greg Blekherman, Mario Kummer, Cordian Riener, and Markus Schweighofer), Annales Henri Lebesgue 3 (2020), pp. 1327--1341.
• Semi-inverted linear spaces and an analogue of the broken circuit complex (with Georgy Scholten), Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 645-661.
• The Chow form of a reciprocal linear space (with Mario Kummer), Michigan Mathematical Journal 68 (2019), no. 4, pp. 831--858.
• Gram spectrahedra (with Lynn Chua, Daniel Plaumann, Rainer Sinn), Ordered algebraic structures and related topics, Contemp. Math., 697, (2017) pp. 81 - 105.
• Low-rank sum-of-squares representations on varieties of minimal degree (with Greg Blekherman, Daniel Plaumann, Rainer Sinn), International Mathematics Research Notices, (2019) no. 1, pp. 33--54.
• Computing complex and real tropical curves using monodromy (with Danielle Brake and Jonathan Hauenstein), Journal of Pure and Applied Algebra 223 (2019), no. 12, 5232–5250.
• Computing Hermitian determinantal representations of hyperbolic curves (with Daniel Plaumann, Rainer Sinn, and David Speyer), International Journal of Algebra and Combinatorics, 25(8) (2015) pp. 1327 - 1336.
  
• A small frame and a certificate of its injectivity. Sampling Theory and Applications (SampTA) Conference Proceedings. (2015), pp. 197 - 200. Supplementary material.
  
• A real stable extension of the Vámos matroid polynomial (with Sam Burton and Yewon Youm).
  
• What is a spectrahedron? Notices of the American Mathematical Society 61(5) (2014) pp. 492 - 494.
  
• An algebraic characterization of injectivity in phase retrieval (with Aldo Conca, Dan Edidin, and Milena Hering), Applied and Computational Harmonic Analysis 38:2 (2015) pp. 346-356.
  
• Quartic spectrahedra (with John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels), Mathematical Programming, 2(151) Series B, pp. 585-612.
  
• Hyperbolic polynomials, interlacers, and sums of squares (with Mario Kummer and Daniel Plaumann), Mathematical Programming, 1(153) Series B (2015), pp. 223-245.
  
• Determinantal representations of hyperbolic plane curves: An elementary approach (with Daniel Plaumann), Journal of Symbolic Computation 57 (2013) pp. 48-60.
  
• The entropic discriminant (with Raman Sanyal and Bernd Sturmfels), Advances in Mathematics, 244 (2013) pp. 678-707.
  
• The central curve in linear programming (with Jesús De Loera and Bernd Sturmfels), Foundations of Computational Mathematics 12 (2012) pp. 509-540.
  
• Computing Linear Matrix Representations of Helton-Vinnikov Curves, with Daniel Plaumann and Bernd Sturmfels), Mathematical Methods in Systems, Optimization and Control, (eds. Harry Dym, Mauricio de Oliveira, Mihai Putinar), Operator Theory: Advances and Applications, Vol 222, Birkhauser, Basel, 2012, pp. 259-277. Supplementary material.
  
• Quartic curves and their bitangents (with Daniel Plaumann and Bernd Sturmfels), Journal of Symbolic Computation 46 (2011) pp. 712-733. Supplementary material.
  
• Edges of the Barvinok-Novik orbitope, Discrete & Computational Geometry 46(33) (2011) pp. 479-487.
  
• Real radical initial ideals, Journal of Algebra, 352(1) (2012), pp. 392-407
  
• Lower bounds for optimal alignments of binary sequences, Discrete Applied Math. 157:15 (2009), pp. 3341-3346.
  
• Mathematical approaches to the pure parsimony problem (with Paul Blain, Courtney Davis, Al Holder, and Jorge Silva), appearing as "Diversity Graphs" in "Clustering Challenges in Biological Networks"

Slides from Talks

• Completely log-concave polynomials and matroids - January 2019
• Geometry of spectrahedra - January 2019
• Spectrahedra - October 2017
• Convex Algebraic Geometry - January 2017
• Low-rank sums-of-squares representations - January 2017
• Reciprocal linear spaces and their Chow forms - January 2016
• Numerical methods for computing real and complex tropical curves - December 2015
• Injectivity of Hermitian frame measurements - July 2015
• Quartic symmetroids and spectrahedra - April 2014
• An algebraic approach to phase retrieval - October 2013
• Computing Hermitian determinantal representations of plane curves - August 2013
• Hyperbolic polynomials, interlacers, and sums of squares - August 2013
• A sums of squares relaxation for hyperbolicity cones - July 2012
• The central curve in linear programming - January 2011
• The convex hull of a parametrized curve - July 2010
• Faces of the Barvinok-Novik Orbitope - February 2010
• Degree stability of SOS modulo an ideal - May 2009
• Parametric sequence alignment - September 2008