Research Interests:
Lie theory, algebraic geometry, representation theory, integrable systems, Poisson geometry.
Grant Support:
NSA, Mathematical Sciences Program, Young Investigator Award (January 2016-December 2017).
Funded Grant Proposal, Fall 2014
Research Statement:
Publications:
The Gelfand-Zeitlin Integrable System and Its Action on Generic Elements of gl(n) and so(n) , New Developments in Lie Theory and Geometry (Cruz Chica, Cordoba, Argentina, 2007), Contemp. Math., vol. 491, Amer. Math. Soc., Providence, RI, 2009, pp.255-281.
The orbit structure of the Gelfand-Zeitlin group on n\times n Matrices , Pacific Journal of Math. 250, (2011), no.1, 109-138.
On algebraic integrability of Gelfand-Zeitlin fields , coauthored with Sam Evens, Transformation Groups, 15 (2010), no. 1, 46-71.
K-oribts on the flag variety and strongly regular nilpotent matrices , coauthored with Sam Evens, Selecta. Math. (N.S.), {\bf 18} (2012), no.1, 159-177.
The Gelfand-Zeitlin integrable system and K-orbits on the flag variety , coauthored with Sam Evens, “Symmetry: Representation Theory and its Applications,” 85-119, Progr. Math., {\bf 257}, Birkauser/Springer, New York, 2014.
Eigenvalue Coincidences and K-orbits, I, coauthored with Sam Evens, Journal of Algebra, {\bf 422} (2015), 611-632.
Lie-Poisson theory for direct limit Lie algebras, coauthored with Michael Lau, J. Pure and Appl. Algebra, {\bf 220}, (2016), no 4, 1489-1516.
Preprints:
Eigenvalue coincidences and multiplicity free spherical pairs, arxiv 1410.3901, 38 pages, coauthored with Sam Evens.
Work soon to be completed:
The geometry and combinatorics of $K$-orbits on the flag variety for multiplicity free spherical pairs (G,K) (joint with Sam Evens) approx 30 pages.
On complex Gelfand-Zeitlin integrable systems (joint with Sam Evens) approx 60 pages.
Other work in progress:
Geometric construction of generalized Harish-Chandra modules for the partial Gelfand-Zeitlin algebra (joint with Sam Evens).
A nonlinear Gelfand-Zeitlin system and a complex Ginzburg-Weinstein diffeomorphism (joint with Sam Evens).
A generalization of the Joseph-Letzter theorem for the Gelfand-Zeitlin algebra of U_{q}(gl(n,C)) (joint with Sam Evens).
Ph D Thesis:
The Gelfand-Zeitlin algebra and polarizations of regular adjoint orbits for classical groups, UC San Diego, 2007.
Supervisor: Nolan Wallach