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.
- 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,
- 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.
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.
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.