RESEARCH INTERESTS

My research interests fit broadly into the area of mathematical physics.  In particular, during my PhD study, I have been focused on Random Matrix Theory, representation theory and their applications in physics.  Random Matrix theory is a unifying method in mathematics and physics.  I have worked on the following problems:

• Random Matrix and Moduli space

One of the major impact of string theory to mathematics is concerned with the moduli space of Riemann surfaces, since a conformal field theory can be represented as integrals on moduli space, which is a finite dimensional space.Based on a 2-D topological gravity theory, E. Witten proposed a conjecture in 1990 that the partition function with infinite many coupling constants satisfies KdV hierarchies, which are a series of nonlinear PDEs.  This conjecture is interesting because it is concerned with the KdV hierarchy, which is equivalent to the annihilations of Virosoro algebras.  Since Virosoro algebras play a role in many models physics, such as the 2-D gravity and 3 dimensional Chern-Simon theories. Kontsevich's proof of Witten's conjecture has shown that the importance of Random Matrix theory and planar graph techniques in moduli space. A general one matrix model is equivalent to a planar graph theory. With Prof. Jorgensen, we conjecture that the nonlinear sigma model for moduli space of holomorphic maps from Riemann surface to a target Kahler manifold can be described by a generalized "planar graph" theory, such that we can get the Virasoro constraints based on this conjectural planar graph theory.

• The Semicircle Law of Fixed Trace Ensemble

E.P.Wigner in 1950s showed that the level density of "border matrix" ensemble which is a kind of real symmetric matrix ensemble, asymptotically approaches to the semicircle.  We would like to consider the Vandermonde ensemble, i.e., the random matrix ensemble which eigenvalues are fixed in a sphere. Different from fixed square ensemble, the eigenvalues of Vandermonde ensemble are fixed in the unit sphere which radius is independent of N.  This assumption brings new difficulty in the spectral analysis.  With Prof. Lihe Wang, we overcome this difficulty by estimates on the integral equation we shall derive, and then prove the semicircle law for Vandermonde ensemble rigorously.  Moreover, it seems that the semicircle law of Vandermonde ensemble can also be derived by combinatorial arguments.  I would like to continue to work on the combinatorial method to prove the semicircle law of Vandermonde ensemble.
 

• The group Integral of Representations of Unitary Groups

The group integrals of unitary groups are important in physics.  The calculation of the entropy will rely on the calculation of group integrals of unitary group.  Moreover, the group integral of unitary groups plays a role in lattice gauge theory.  In Weingarten's work, the author gave the asymptotic behavior of group integrals.  With Professor Jorgensen, we use an invariant polynomial method to get the exact formula to compute group integrals and reobtain the old results.  For group integrals of irreducible representations of unitary groups, we generalize Weyl-Schur duality theorem and give an algorithm to compute the group integrals.
 

• The Index Theorem and Supersymmetry

Fujikawa gave a very direct method to derive the gauge anomaly of a spin field, which explains the topological origin of anomalies most clearly. Gravitational anomaly was derived by Witten using Fujikawa's method.  Alvarez-Gaumé and Witten pointed out the relationship between gravitational anomalies and the index theorem.   We derive the Hirzebruch signature theorem and the Hirzebruch-Riemann-Roch theorem in detail.   In the derivation of the Hirzebruch-Riemann-Roch theorem, we need to consider a supersymmetric theory coupled with gauge field.  Some papers got the index of twisted spin field (twisted by gauge field) using very complicated calculation.  However we will give a very simple idea from a physical point of view, which can be applied to any gauge coupled index, to overcome this difficulty.  I would like to see the connection between index theorem, supersymmetry and number theory in the future.  

• Philosophical and historical aspects of mathematical physics

This is my minor interest and I fully understand to think about philosophical issue, one need a lot of mathematical practice. I hope I will work on some problems in this field in the future of my career.