Greg Blekherman



 


Greg Blekherman
Professor 
Georgia Tech

Email: greg@math.gatech.edu

          


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.


SIAM Journal on Applied Algebra and Geometry

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



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

Together with Annie Raymond and Rekha Thomas I am organizing a two week summer school, July 13-July 24th 2020, at MSRI. Further information is here.

The Summer School is postponed until Summer 2022 due to COVID-19.


Papers and Preprints

Combinatorics

  •  Proof of the Erdős-Simonovits conjecture on walks”, G. Blekherman, A, Raymond, arXiv:2009.10845.
  • Tropicalization of Graph Profiles”, G. Blekherman, A, Raymond, M. Singh, R. Thomas, arXiv:2004.05207, submitted for publication.
  • “Threshold Graphs Maximize Homomorphism Densities”, G. Blekherman, S. Patel, arXiv:2002.12117, submitted for publication.
  • Bounds on Regularity of Quadratic Monomial Ideals"G. Blekherman, J. Jung, arXiv:1906.06358, Journal of Combinatorial Theory, Series A (2020).
  • Simple Graph Density Inequalities with no Sum of Squares Proofs", G. Blekherman, A, Raymond, M. SinghR. Thomas, arXiv:1812.08820Combinatorica (2020).
Real Algebraic Geometry and Optimization
  • Sums of Squares: A Real Projective Story",  G. Blekherman, R. Sinn, G. Smith, M. Velasco, an expository paper written for the Notices of the AMS, arXiv:2101.05773

  • “Hyperbolic Relaxation of k-Locally Positive Semidefinite Matrices”, G. Blekherman, S. Dey, K. Shu, S. Sun, arXiv:2012.04031submitted for publication.
  • “Sums of Squares and Sparse Semidefinite Programming”, G. Blekherman, K. Shu, arXiv:2010.11311 submitted for publication.
    • “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:2002.02988Math Programming (2020).
    • Sums of Squares and Quadratic Persistence on Real Projective Varieties",  G. Blekherman, R. Sinn, G. Smith, M. Velasco,  arXiv:1902.02754accepted in Journal of the European Math Society (JEMS).

    • Symmetric Nonnegative Forms and Sums of Squares, G. Blekherman, C.Riener,  arXiv:1205.3102Discrete 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 Noticesfinal 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:1406.1873Journal 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 AMS25, 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:1909.06593Journal of Pure and Applied Algebra (2020).
    • “Typical and Generic Ranks in Matrix Completion, D. Bernstein, G. Blekherman, R. Sinn, arXiv:1802.09513Linear 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:1511.07044Linear 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 (2020).
    • Generalized Eigenvalue Methods for Gaussian Quadrature Rules", G. BlekhermanM. KummerC. 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
    • 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:2012.04855submitted for publication.

    • “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.




    Summer School on Hyperbolic Polynomials, Sums of Squares and Optimization

    I am organizing s Summer School on Hyperbolic Polynomials, Sums of Squares and Optimization at Georgia Tech, June 25-29, 2018. Main Lecturers are Daniel Plaumann (TU Dortmund), Rainer Sinn (FU Berlin), Cynthia Vinzant (NC State) and Greg Blekherman (Georgia Tech). Topics covered are:
    • Geometry of Hyperbolic Polynomials and Sums of Squares
    • Conic and Hyperbolic Programming
    • Interlacing Polynomials
    • Stable Polynomials in Combinatorics
    • Sums of Squares in Combinatorics and Optimization

    Please visit the program homepage for more information. We have NSF funding for participants. Please visit Applications to apply.


    Summer School on Real Algebraic Geometry and Optimization, July 11-15, 2016.

    Rainer Sinn (Georgia Tech), Mauricio Velasco (Universidad de los Andes) and I organized a summer school at Georgia Tech. The topics covered will be:
    • Geometry of Sums of Squares and Nonnegative Polynomials
    • Applications of Sums of Squares Methods in Optimization and Engineering

    Please visit program homepage for more information and applicat


    Teaching, Spring 2017.

    The course page for Honors Calculus III is here.

    The page for Linear Programming is here.



    Real Algebraic Geometry and Optimization, Fall 2015


    The link to course resources is 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.


    SIAM Conference on Applied Algebraic Geometry

    The conference took place in Daejeon, Korea, August 3-7th 2015.
    Here is the conference homepage. 
    The conference was supported by NSF travel grant DMS-1522597. 


    2012 IMA Summer Program in Algebraic Geometry for Applications

    Together with Anton Leykin and Josephine Yu I organized 3 week IMA sponsored graduate student summer program in Algebraic Geometry for Applications. The dates for the program are June 18--July 6th 2012. 

    Topics covered were: Polynomial Optimization and Real Algebraic Geometry, Computer Algebra and Numerical Algebraic Geometry, and Tropical Geometry. Further information about the program can be found here.




    Support

    My research is supported by NSF CAREER award DMS-1352073. I was previously supported by a Sloan Research Fellowship.




    Recent Talks (to be updated)
    • Combinatorics Seminar, University of Washington, January 2013.
    • Algebraic Geometry Seminar, University of Chicago, January 2012.
    • Algebraic Geometry Seminar, University of Michigan, October 2012.
    • Tensors and their Geometry in High Dimensions, UC Berkeley, September 2012.
    • MAA Invited Paper Session on Convex Algebraic Geometry, August 2012.
    • SIAM Annual Meeting, Session on Convex Algebraic Geometry and Optimization, July 2012
    • MAA Invited Paper Session on Sum of Squares Polynomials in Optimization, Joint AMS-MAA Meetings, January 2012.
    • Optimization Seminar, ISYE, Georgia Tech, December 2011.
    • Algebraic Geometry Seminar, Queens University, October 2011
    • SIAM Conference on Applied Algebraic Geometry. Session on Algebraic Geometry in Convex Optimization, October 2011.

    • Algebraic Geometry Seminar, Emory University, September 2011.
    • Algebraic Geometry Seminar, University of Georgia, September 2011.
    • Combinatorial Methods in Algebraic Geometry and Commutative Algebra, Max Planck Institute, July 2011.
    • Applied Math Seminar. Institut de Mathématiques de Toulouse, June 2011.
    • SIAM Optimization Meeting. Session on Algebraic Geometry and Optimization. May 2011.

    • Algebraic Geometry Seminar. Università di Firenze. May 2001.

    • Long Program Seminar. Mittag-Leffler Institute. May 2011.

    • Workshop on Applications of Schubert Calculus. University of Iowa. March 2011.

    • Center for Computational Mathematics Seminar, UC San Diego, February 2011.

    • Algebraic Geometry and Commutative Algebra Seminar, UC Berkeley, December 2010.
    • Convex Optimization and Algebraic Geometry Workshop, IPAM, September 2010.

    • SIAM Annual Meeting, Session on Convex Algebraic Geometry, July 2010.

    • MTNS 2010, Session on Real Algebraic Geometry and Applications, July 2010.

    • Symbolic Computation Seminar, North Carolina State University, March 2010.

    • Workshop on Randomization, Relaxation, and Complexity, Banff International Research Station, February 2010.

    • Workshop on Convex Algebraic Geometry, Banff International Research Station, February 2010.

    • Math Department Colloquium, UC San Diego, October 2009.

    • Meeting of Focused Research Group on Semidefinite Optimization and Convex Algebraic Geometry, MIT, May 2009.

    • Math Department Colloquium. Hobart and William Smith Colleges, April 2009.

    • AMS Sectional Meeting, Session on Concrete Aspects of Positive Polynomials, Urbana-Champaign, March 2009.

    • Geometry Seminar, The Ohio State University, January 2009.


    Education


    Contact

    School of Mathematics
    Georgia Tech
    686 Cherry Street
    Atlanta, GA 30332

    greg@math.gatech.edu