Research

Research interests:

My research revolves around noncommutative generalizations of real algebra and real algebraic geometry, including:

• ordered algebras and algebras with involution,
• representation theory,
• obstructions to orderability,
• semidefinite programming,
• quadratic and hermitian forms, forms of higher level,
• noncommutative valuation theory.

Published papers

Preprints

• Real algebraic geometry for matrices over commutative rings, Arxiv , submitted
• (with Bill Helton, Scott McCullough, Christopher Nelson) A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms, Arxiv , submitted
• (with Yurii Savchuk and Konrad Schmüdgen) On q-normal operators and quantum complex plane, Arxiv , To appear in Transactions of the AMS.
• On generalized real semiprime ideals, pdf, submitted.

Slides of my talks

Visitors

Editorial work

• European Journal of Pure and Applied Mathematics, associate editor
• Journal of Algebraic Statistics, editor
• Journal of Algebra and Computational Applications, editor

Interesting meetings

• 2012
  • Emerging Developments in Real Algebraic Geometry: Positivity, Convexity, NC-Geometry, Optimization , Magdeburg February 20-24, 2012.
  • Algebra and Geometry, International Conference dedicated to the 65-th anniversary of Askold G. Khovanskii , Moscow, Russia, June 4 - 9, 2012
  • 2012 IMA PI Summer Program for Graduate Students Algebraic Geometry for Applications , Georgia Tech, June 18th - July 6th.
  • Summer school on semidefinite optimization September 3-7, 2012.
• 2011
  • Real Algebra, Geometry and Convexity , Leipzig (Germany), February 14-18, 2011.
  • Real Algebraic Geometry Conference, Rennes (France), June 20-24 2011.
  • 3rd International Conference on Matrix Methods in Mathematics and Applications, MMMA-2011, Moscow (Russia), June 22nd-25th, 2011,
  • IWOTA 2011 Universidad de Sevilla (Spain), July 3rd-9th, 2011,
  • The Second International Conference and Workshop on Valuation Theory Segovia / El Escorial (Spain), July 18th - 29th, 2011.
• 2010
  • Mini Course 2010: Linear Algebraic Groups and Quadratic Forms, Patching, Involutions and Buildings Lens (France), June 14-18 2010,
  • 9th Colloquiumfest on Algebra and Logic & 8th Czech, Polish and Slovak Conference on Number Theory, Bukowina Tatrzańska (Poland), June 21—24, 2010
  • The conference "Functions and Operators" (FaO2010) (a conference dedicated to Franciszek Hugon Szafraniec's 70th birthday), Krakow (Poland), June 21-25, 2010.
  • Algebraic Geometry, K-theory, and Motives (a conference dedicated to Andrei Suslin's 60th birthday), St. Petersburg (Russia), June 25-29, 2010.
  • The 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), Budapest (Hungary), July 5-9, 2010.
  • The 21st International Workshop on Operator Theory and its Applications (IWOTA 2010), Berlin (Germany), July 12-16, 2010.
  • Convex Optimization and Algebraic Geometry, Los Angeles (USA), September 28 - October 1, 2010.

Research statement