Greg Blekherman



 


Greg Blekherman
Associate Professor 
Georgia Tech

Email: greg@math.gatech.edu

          


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

Rainer Sinn (Georgia Tech), Mauricio Velasco (Universidad de los Andes) and I are organizing 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 applications.


 

Research Interests

  • Applied Algebraic Geometry

  • Convex Geometry

  • Optimization

  •  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.


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.


Papers and Preprints

  • “Do Sums of Squares Dream of Free Resolutions?", G. Blekherman, R. Sinn, M. Velasco,  arXiv:1607.03551, submitted for publication.
  • “Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree", G. Blekherman, D. Plaumann, R. Sinn, C. Vinzant, arXiv:1606.04387, submitted for publication.
  • “Sharp Degree Bounds for Sum-of-Squares Certificates on Projective Curves", G. Blekherman, G. Smith, M. Velasco,  arXiv:1605.08330, submitted for publication.
  • On real typical ranks”, A. Bernardi, G. Blekherman, G. Ottaviani, arXiv:1512.01853, submitted for publication.
  • Real Rank with Respect to Varieties”, G. Blekherman, R. Sinn, arXiv:1511.07044, Linear Algebra and Its Applications, 505, 344-360 (2016).
  • Extreme Rays of the Hankel Spectrahedra for Ternary Forms”, G. Blekherman, R. Sinn, arXiv:1406.1873, to appear in Journal of Symbolic Computation.
  • “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, DOI 10.1007/s00209-016-1644-7, arXiv:1402.4199.
  • "On Maximum, Typical, and Generic Ranks", G. Blekherman, Z. Teitler, Mathematische Annalen, 362(3), 1021-1031, (2015) arXiv:1402.2371.
  • "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 2015(18), 8427-8470, (2015) arXiv:1305.0642.
  • "Typical Real Ranks of Binary Forms", G. Blekherman, Foundations of Computational Math, 15(3), 793-798 (2015), arXiv:1205.3257.
  • "Symmetric Nonnegative Forms and Sums of Squares", G. Blekherman, C. Riener,  arXiv:1205.3102, in revision.

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

  • "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.
  • "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.

  • “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.
  • “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.

  • “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.

  • “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.

  • “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.


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