My research interest lies in the interaction between geometric representation theory and
symplectic/Poisson geometry. The problems that I think about usually have to do with
deformation quantization, cluster algebras, spherical varieties, (additive) toric varieties, gauge theory, mirror symmetry, etc.
In particular, I am interested in anything that is even remotely related to the
standard Poisson
bracket on a semisimple Lie group. Examples of such objects include Lie bialgebra, Manin triple,
variety of Lagrangian subalgebras, Fomin-Zelevinsky minor, universal centralizer, quantum cohomology,
Yang-Baxter equation, quantized universal enveloping algebra, crystal base, quantum boson algebra, abelian
ideal, Kazhdan-Lusztig theory for an affine Weyl group, etc.

Up to now, the theory of abelian ideals
of a fixed Borel subalgebra of a simple Lie algebra has been dominantly combinatorial. Currently, I am
developing a Poisson geometric approach to this theory. I am also trying to develop a theory of additive
toric varieties. A prototypical example of such a variety is the wonderful compactification of a Cartan
subalgebra of a semisimple Lie algebra.

My research statement is available upon request.

Papers

Y. Li: Wonderful compactification of a Cartan subalgebra of a semisimple Lie algebra. In preparation, to be posted on the arXiv very soon.