## Research Interests

Applied Algebraic Geometry

Convex Geometry

Combinatorics

Optimization

Mathematical Biology

My work focuses on the interplay between convex and algebraic geometry (both real and complex), especially in the area of sums of squares approximations to nonnegative polynomials, and tensor ranks and their generalizations.

## Combinatorics at Georgia Tech

Georgia Tech has thriving research groups in all areas of combinatorics, including probabilistic, algebraic, extremal, and additive combinatorics, and graph theory. See more here.

## Semidefinite Optimization and Convex Algebraic Geometry

Pablo Parrilo, Rekha Thomas and I edited "Semidefinite Optimization and Convex Algebraic Geometry", which is now out as Volume 13 in SIAM Optimization Series. You can download a free pdf version of the book here.

## Algorithms, Combinatorics and Optimization (ACO)

I am an associate director of the Algorithms, Combinatorics and Optimization (ACO) program at Georgia Tech.

SIAM Activity Group on Algebraic Geometry SI(AG)2

I served as the chair of the SIAM Activity Group on Algebraic Geometry from 2021-2023. Please join us for the SIAM AG25 Meeting!

## SIAM Journal on Applied Algebra and Geometry

I am an associate editor for SIAM Journal on Applied Algebra and Geometry.

## Southeast Center for Mathematical Biology

I am part of the NSF-Simons Southeast Center for Mathematical Biology. Together with Dan Goldman we are working on understanding principles of animal locomotion and applications in robotics.

## BIRS-MSRI Summer School on Sums of Squares Method in Geometry, Combinatorics and Optimization

Together with Annie Raymond and Cynthia Vinzant I organized a two week summer school, July 31-August 12 2022. Further information is here. Problem sets from the first week are below.

Greg Problem Set 1 is here.

Greg Problem Set 2 is here.

Greg Problem Set 3 is here.

## Papers and Preprints

Combinatorics

“Ubiquity of Power Sums in Graph Profiles", G. Blekherman, A. Raymond, arXiv:2308.07422, submitted for publication.

“Undecidability of polynomial inequalities in weighted graph homomorphism densities”, G. Blekherman, A. Raymond, F. Wei, Forum of Math, Sigma (2024), final version here.

“A Path Forward: Tropicalization in Extremal Combinatorics”, G. Blekherman, A. Raymond, arXiv:2108.06377, Advances in Math 407(8) (2022).

“Proof of the Erdős-Simonovits conjecture on walks”, G. Blekherman, A. Raymond, arXiv:2009.10845, accepted in Graphs and Combinatorics.

“Tropicalization of Graph Profiles”, G. Blekherman, A. Raymond, M. Singh, R. Thomas, arXiv:2004.05207, Transactions of the AMS (2022).

“Threshold Graphs Maximize Homomorphism Densities”, G. Blekherman, S. Patel, accepted in Combinatorics, Probability and Computing (2023), final version here.

“Bounds on Regularity of Quadratic Monomial Ideals", G. Blekherman, J. Jung, arXiv:1906.06358, Journal of Combinatorial Theory, Series A, 176 (2020).

“Simple Graph Density Inequalities with no Sum of Squares Proofs", G. Blekherman, A, Raymond, M. Singh, R. Thomas, arXiv:1812.08820, Combinatorica (2020).

Real Algebraic Geometry and Optimization

“Nonnegativity certificates on real algebraic surfaces", G. Blekherman, R. Sinn, G. Smith, M. Velasco, arXiv:2409.08834, submitted for publication.

“Symmetric nonnegative functions, the tropical Vandermonde cell and superdominance of power sums", J. Acevedo, G. Blekherman, S. Debus, C. Riener, arXiv:2408.04616, submitted for publication.

“On odd powers of nonnegative polynomials that are not sums of squares", G. Blekherman, K. Kozhasov, B. Reznick, arXiv:2407.21779, submitted for publication.

“Nonnegative Polynomials and Moment Problems on Algebraic Curves", L. Baldi, G. Blekherman, R. Sinn, arXiv:2407.06017, submitted for publication.

“A Topological Approach to Simple Descriptions of Convex Hulls of Sets Defined by Three Quadrics", G. Blekherman, A. Dunbar, arXiv:2405.18282, submitted for publication.

“K-Lorentzian Polynomials", G. Blekherman, P. Dey, arXiv:2405.12973, submitted for publication.

“Convex Ternary Quartics are SOS-Convex", A.A. Ahmadi, G. Blekherman, P. Parrilo, arXiv:2404.14440, submitted for publication.

“Symmetric Hyperbolic Polynomials”, G. Blekherman, J. Lindberg, K. Shu, arXiv:2308.09653, submitted for publication.

“Power Mean Inequalities and Sums of Squares”, J. Acevedo, G. Blekherman, arXiv:2303.11823, accepted in Discrete and Computational Geometry.

“The Wonderful Geometry of the Vandermonde map”, J. Acevedo, G. Blekherman, S. Debus, C. Riener, arXiv:2303.09512, submitted for publication.

“Aggregations of quadratic inequalities and hidden hyperplane convexity”, G. Blekherman, S. Dey, S. Sun, arXiv:2210.01722, accepted in SIAM Journal on Optimization.

“Weak and Strong Extremal Biquadratics", G. Blekherman, Bogdan Raiţă, Isabelle Shankar, Rainer Sinn, arXiv:2204.10625.

“Moments, Sums of Squares, and Tropicalization", G. Blekherman, F. Rincon, R. Sinn, C. Vinzant, J. Yu, arXiv:2203.06291, submitted for publication

“Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment", G. Blekherman, M. Kummer, R. Sanyal, K. Shu, S. Sun, International Math Research Notices (2023), final version here.

“Sums of squares, Hankel index, and almost real rank", G. Blekherman, J. Chen, J. Jung, arXiv:2108.06635, accepted in Forum of Math, Sigma.

“Lectures on Nonnegative Polynomials and Sums of Squares", G. Blekherman, J. Wesner, arXiv:2105.14315

“Sums of Squares: A Real Projective Story", G. Blekherman, R. Sinn, G. Smith, M. Velasco, an expository paper which appeared in the Notices of the AMS (2021), arXiv:2101.05773

“Hyperbolic Relaxation of k-Locally Positive Semidefinite Matrices”, G. Blekherman, S. Dey, K. Shu, S. Sun, arXiv:2012.04031, SIAM Journal on Optimization 22(2), (2022).

“Sums of Squares and Sparse Semidefinite Programming”, G. Blekherman, K. Shu, arXiv:2010.11311, SIAM Journal on Applied Algebra and Geometry, 5(4), (2021).

“A Brief Introduction to Sums of Squares”, G. Blekherman, short introductory article based on the AMS Sums of Squares short course, Proceedings of Symposia in Applied Mathematics, AMS.

“Sparse PSD approximation of the PSD cone”, G. Blekherman, S. Dey, M.Molinaro, S. Sun, arXiv:2002.02988, Math Programming, 191, (2022).

“Sums of Squares and Quadratic Persistence on Real Projective Varieties", G. Blekherman, R. Sinn, G. Smith, M. Velasco, arXiv:1902.02754, Journal of the European Math Society (JEMS) (2021).

“Symmetric Nonnegative Forms and Sums of Squares”, G. Blekherman, C.Riener, arXiv:1205.3102, Discrete and Computational Geometry (2020).

“Do Sums of Squares Dream of Free Resolutions?", G. Blekherman, R. Sinn, M. Velasco, SIAM Journal on Applied Algebra and Geometry, 1(1), 175–199, final version here, (2017).

“Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree", G. Blekherman, D. Plaumann, R. Sinn, C. Vinzant, International Math Research Notices, final version here, (2017).

“Sharp Degree Bounds for Sum-of-Squares Certificates on Projective Curves", G. Blekherman, G. Smith, M. Velasco, arXiv:1605.08330, Journal de Mathématiques Pures et Appliquées (2019).

“Extreme Rays of the Hankel Spectrahedra for Ternary Forms”, G. Blekherman, R. Sinn, arXiv:1406.1873, Journal of Symbolic Computation (2017).

“Gap Vectors of Real Projective Varieties”, G. Blekherman, S. Iliman, M. Juhnke-Kubitzke, M. Velasco, Advances in Math, 183, 458-472, (2015) arXiv:1407.0585.

“Sums of Squares on the Hypercube”, G. Blekherman, J. Gouveia, J. Pfeiffer, Mathematische Zeitschrift, 284(1-2), 41–54, (2016) arXiv:1402.4199.

“Sums of Squares and Varieties of Minimal Degree", G. Blekherman, G. Smith, M. Velasco, Journal of the AMS, 29, 893-913, (2016), arXiv:1308.0751.

“Dimensional Differences Between Faces of the Cones of Nonnegative Polynomials and Sums of Squares", G. Blekherman, S. Iliman, M. Kubitzke, International Math Research Notices (18), 8427-8470, (2015) arXiv:1305.0642.

“Positive Gorenstein Ideals", G. Blekherman, Proceedings of the AMS, 143, 69-86, (2015), arXiv:1203.3775.

"Algebraic Boundaries of Hilbert's SOS Cones", G. Blekherman, J. Hauenstein, J. C. Ottem, K. Ranestad, B. Sturmfels, Compositio Mathematica, Vol. 148(6), 2012, 1717-1735, arXiv:1107.1846.

“Nonnegative Polynomials and Sums of Squares”, G. Blekherman, Journal of the AMS, 25, 2012, 617-635 , arXiv:1010.3465.

“Dimensional Differences between Nonnegative Polynomials and Sums of Squares”, G. Blekherman, arXiv:0907.1339.

“Convex Forms that are not Sums of Squares”, G. Blekherman, arXiv:0910.0656.

“There are Significantly More Nonnegative Polynomials than Sums of Squares”, G. Blekherman, Israel Journal of Mathematics, Vol. 153, 2006 arXiv:0402.5158.

“Convex Geometry of Orbits”, A. Barvinok and G. Blekherman, Combinatorial and Computational Geometry, MSRI Publications, Vol. 52, 2005 arXiv:0312.5268.

“Convexity Properties of The Cone of Nonnegative Polynomials”, G. Blekherman, Discrete and Computational Geometry, Vol. 32, no 3, 2004 arXiv:0211.5176.

Tensor Rank and Matrix Completion

“Typical Ranks in Symmetric Matrix Completion”, D. Bernstein, G. Blekherman, K. Lee, arXiv:1909.06593, Journal of Pure and Applied Algebra, 225(7), (2021).

“Typical and Generic Ranks in Matrix Completion”, D. Bernstein, G. Blekherman, R. Sinn, arXiv:1802.09513, Linear Algebra and Its Applications (2020).

“Maximum Likelihood Threshold and Generic Completion Rank of Graphs", G. Blekherman, R. Sinn, arXiv:1703.07849, Discrete and Computational Geometry, 61(2), 303-324 (2018).

“On Real Typical Ranks”, A. Bernardi, G. Blekherman, G. Ottaviani, arXiv:1512.01853, Bollettino dell'Unione Matematica Italiana (2017).

“Real Rank with Respect to Varieties”, G. Blekherman, R. Sinn, arXiv:1511.07044, Linear Algebra and Its Applications, 505, 344-360 (2016).

"On Maximum, Typical, and Generic Ranks", G. Blekherman, Z. Teitler, Mathematische Annalen, 362(3), 1021-1031, (2015) arXiv:1402.2371.

"Typical Real Ranks of Binary Forms", G. Blekherman, Foundations of Computational Math, 15(3), 793-798 (2015), arXiv:1205.3257.

Moment Problem

“Quantum Entanglement, Symmetric Quadratic Nonnegative Polynomials and Moment Problems", G. Blekherman, H.M. Bharath, arXiv:1904.00072, Math Programming, 193, (2020).

“Generalized Eigenvalue Methods for Gaussian Quadrature Rules", G. Blekherman, M. Kummer, C. Riener, M. Schweighofer, C. Vinzant, arXiv:1805.12047, Annales Henri Lebesgue (2020).

“The Core Variety and Representing Measures in the Truncated Moment Problem", G. Blekherman, L. Fialkow, arXiv:1804.04276, Journal of Operator Theory (2020).

"The Truncated K-Moment Problem for Closure of Open Sets", G. Blekherman, J. B. Lasserre, Journal of Functional Analysis, Vol. 263(11) (2012), 3604–3616 arXiv:1108.0627.

Math Biology

“Optimizing contact patterns for robot locomotion via geometric mechanics”, B. Chong, T. Wang, B. Lin, S. Li, P. Muthukrishnan, J. He, G. Blekherman, H. Choset, D. I. Goldman, accepted in International Journal of Robotics Research.

“Moving sidewinding forward: optimizing contact patterns for limbless robots via geometric mechanics”, B. Chong, T. Wang, B. Lin, S. Li, H. Choset, G. Blekherman, D. I. Goldman; Finalist for Best Paper award at RSS 2021.

“Frequency Modulation of Body Waves to Improve Performance of Sidewinding Robots”, B. Chong, T. Wang, J. M. Rieser, B. Lin, A. Kaba, G. Blekherman, H. Choset, D. I. Goldman, accepted in International Journal of Robotics Research.

“Reconstruction of Backbone Curves for Snake Robots”, T. Wang, B. Lin, B. Chong, J. Whitman, M. Travers, D. I. Goldman, G. Blekherman, H. Choset, arXiv:2012.04855, IEEE Robotics and Automation Letters.

“Optimizing coordinate choice for locomotion systems with toroidal shape spaces”, B. Lin, B. Chong, Y. Ozkan-Aydin, E. Aydin, H. Choset, D. I. Goldman, G. Blekherman, Proceedings of IROS 2020.

“Bioinformatics Tools for Cancer Metabolomics”, G. Blekherman, R. Laubenbacher, D. Cortes, P. Mendes, F. Torti, S. Akman, S. Torti and V. Shulaev, Metabolomics, Vol. 7, no. 3, 2011.

"ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra", F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, G. Blekherman, A. Veliz-Cuba and R. Laubenbacher, BMC Bioinformatics Vol. 12, no. 1, 2011 arXiv:1012.0778.

“Differential Gene Expression in Normal and Transformed Human Mammary Epithelial Cells in Response to Oxidative Stress”, D. Cortes, W. Sha, V. Hower, G. Blekherman, R. Laubenbacher, S. Akman, S. Torti and V. Shulaev, Free Radical Biology and Medicine, Vol. 50(11), no. 1, 2011, 1565-1574.

Miscellaneous

“Experimental Study of Energy-Minimizing Point Configurations on Spheres”, B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly, A. Schürmann, Experimental Mathematics, Vol. 18, no 3, 2009 arXiv:0611.5451.

“The Polyhedral Tammes Problem”, K. Bezdek, G. Blekherman, R. Connelly and B. Csikós, Archiv der Mathematik, Vol. 76, 2001.

“Danzer-Grünbaum’s Theorem Revisited”, K. Bezdek and G. Blekherman, Periodica Mathematica Hungarica, Vol. 39, 1999.

## Education

Ph.D., Mathematics, University of Michigan, 2005

B.A., Mathematics, New York University, 2000

## Contact

School of Mathematics

Georgia Tech

686 Cherry Street

Atlanta, GA 30332