My original research interests were in the Schubert calculus, which is at the interface of algebraic geometry, representation theory and algebraic combinatorics. Over the years, this has evolved in many directions, including real algebraic geometry, computational algebraic geometry, spectral theory, Hopf algebras in combinatorics, discrete and computational geometry, and tropical geometry. Currently (June 2021), I am particularly interested in numerical computation in algebraic geometry and in studying Galois groups in enumerative geometry, and in applications of algebraic geometry to spectral theory. Another interest is to use the theory of Newton-Okounkov bodies in applications of algebraic geometry. I have research projects in most of these areas, and have projects involving over two dozen collaborators.
More information may be found browsing my web page; I particularly recommend
some mathematical short stories or
some more involved web pages that I have created in the
course of my research.
You may also find the descriptions of my research in recent successful grant proposals worthwhile.
|National Science Foundation Standard Grant Combinatorial Algebraic Geometry for Spectral Theory and Galois Groups, 1 August 2022 – 31 July 2025. $303,805. DMS-2201005.|
|Simons Foundation Collaboration Grant for Mathematics Applicable and Combinatorial Algebraic Geometry, 1 September 2019 – 31 August 2022. $25,200. Award Number 636314. proposal.|
|NSF individual research grant Combinatorial and Real Algebraic Geometry, 1 June 2015 -- 31 May 2019. $347,360. DMS-1501370. proposal.|
|NSF individual research grant Applications and Combinatorics in Algebraic Geometry, 1 August 2010 -- 31 July 2014. $235,395. DMS-1001615. proposal.|
|NSF Individual Research grant, Numerical Real Algebraic Geometry, 1 October 2009 -- 31 July 2013. DMS-0915211. proposal.|
|Co-PI with Professor Luis García-Puente of Sam Houston State University on a Texas Advanced Research Projects grant 010366-0054-2007, Algebraic Geometry in Algebraic Statistics and Geometric Modeling, 15 May 2008 -- 14 May 2010. Proposal.|
|NSF Individual Research grant, "Applicable Algebraic Geometry: Real Solutions, Applications, and Combinatorics" 1 September 2007 -- 31 August 2010. DMS-0701050 Proposal.|