Here, you can find my PPT for presentations on research seminars and conferences.
1. U-statistics for determinantal point processes
In this PPT, I will present my recent work with D. Yao on the U-statistics of the determinantal point processes (DPPs). I studied this problem with D.Yao since 2022 on the U-statistics of infinite Ginibre ensemble, which seems a much easier model to work with, but it's not. Our original motivation is to study its random topology inspired by the research on that of Poisson point process conducted by research group of R. Adler - one of my coauthors, where we found the study of its U-statistics is a must. Our first contribution in 2022 is to derive a graph representation for the cumulants of U-statistics of any DPPs, which generalizes Soshnikov's formula on the linear statistics of DPPs. Based on this new graph formula, we derived the first 2 orders of U-statistics for the infinite Ginibre ensemble. The 1st order is the Gaussian limit; when it's degenerate, the 2nd order is a mixture of chi-squared distribution and Gaussian.
In 2023, we studied another model with F. Götze, based on the graph representation again, we discovered that the complete Wiener chaos exists for the spherical case associated with the spectral projection kernel with respect to the Laplace operator. This model seems quite complicated but the main result is much better. When considering the higher order degeneration of the U-statistics, the leading order terms of the cumulants are given by the so-called complete paring graphs in exactly the same pattern as the i.i.d. case. To the best of our knowledge, this is the first result demonstrating the existence of complete Wiener chaos for DPPs.
Click here to download the slides.
2. Extreme gaps for classical random matrices
This is the PPT I gave a seminar talk (online) at Probability Victoria Seminar on random matrices.
In the first part of the talk, I will provide a quick introduction to various classical random matrix models, including the circular ensembles, Gaussian ensembles, and Wishart matrices, along with some classical results on the distribution of their eigenvalues. I will also briefly highlight surprising appearances of random matrix phenomena in several seemingly unrelated areas of mathematics. In the second part, I will present results on extreme gap problems – namely, the smallest and largest spectral gaps – for circular and Gaussian random matrices and propose several conjectures.
Click here to download the slides.
3. A short survey on random polytopes
This is a short survey I gave at Informal Friday Seminar at the University of Sydney. It's a collection of few of the previous results on random polytopes. Suppose we sample random points in a convex body uniformly or Gaussian random vectors in Euclidean space, and look at the polytope they generate. What can I say about it? What is its expected volume, number of vertices, number of edges etc? This talk will provide a short introduction to these kinds of questions. I also introduced several important related concepts and formulas, including Weyl’s tube formula, the Crofton's formula, the FGK formula, and Bourgain’s conjecture on isotropic constants and hyperplane conjecture, etc.
Click here to download the slides.