I studied complex geometry and PDEs during my Ph.D.  After my graduation, I have been working on random matrices, random geometry, and their extensions. Here is a summary of the key findings from my selected publications. 

• The extreme gap problem for eigenvalues of random matrices is classical and fundamental. The Poisson nature of the smallest gaps of determinantal point processes, e.g., CUE and GUE, is understood very well, but there was no previous knowledge beyond the determinantal structure. In our paper (published in GAFA), we finally overcome the determinantal structure, and derived the smallest gaps for GOE which has a Pfaffian structure. The method is based on the observation that the pair of the smallest gaps will stick together, causing the system to become a two-component log-gas in the limit.  All the essential computations turn out to be some type of Selberg Integral. The method can be applied to CβE for integer β (published in AOP). 

• However, for GSE, this method no longer works since the Selberg Integral for the two-component log-gas of GSE is intractable. In our recent preprint, we developed some very brutal-force computations to tackle it, and this systematic method can be used to study the smallest gaps of any Pfaffian point processes with an explicit kernel matrix. 

• The largest gaps between eigenvalues are significantly more challenging compared to the smallest gaps. To prove the Poisson limit of the largest gaps, one has to prove the splitting of the gap probability for relative large scale intervals or domains, which is in fact a type of large deviation result. In our paper, we proved the Poisson limit for the largest gaps of CUE and the interior of GUE based on their determinantal structures. The problem regarding the largest gaps is quite open for other classical ensembles, e.g., COE and CSE. 

• Very recently, in collaboration with Dong Yao, we studied the smallest geodesic distances between zeros of Gaussian random holomorphic sections over Riemann surfaces. The main result is that the smallest geodesic distances tend to a Poisson point process with a universal rate. The result applies specifically to the classical SU(2) random polynomials. Again, the problem regarding the largest geodesic distances is open.  

• The Sachdev-Ye-Kitaev model (SYK) is very topical recently in high energy physics, which is to simply model a black hole. It is also a beautiful quantum spin glass model and a random matrix model.  In our 3 series papers (joint with Dongyi Wei and Gang Tian), we tried to establish some fundamental results from mathematical point of view regarding the distribution of eigenvalues, including the global density, the central limit theorem and the large deviation principle. But more problems are left unsolved, especially the behaviors of the largest eigenvalues, e.g., the growth order, the global distribution and its rescaling limit.

• Last year, within random matrix theory, joint with Dong Yao and Friedrich Götze, we studied the U-statistics (rather than linear statistics) for determinantal point processes. We studied a model case on spheres, and we have uncovered a surprising connection with Wiener chaos. To prove this, we first derived a graphical representation for the cumulant of the U-statistics, which extends the famous Soshnikov's formula to multivariate cases. Our computations can be applied to any determinantal point processes, and Wiener chaos are expected.  

• In random geometry, in collaboration with Robert Adler,  we derived the supremum of the random waves on spheres by Weyl's tube formula (published in AOP). The essential part is to show that there is a lower bound for the critical radius of the embedding of the spheres into higher dimensional space via the spherical harmonics. In the paper under construction, we further proved that such results hold for any Riemannian manifolds, and the critical radius has a universal limit which has its own interest in Riemannian geometry.