Abstract：Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the gap between the first and second largest eigenvalues—measures the expanding property of the graph. In this talk, I will focus on random $d$-regular graphs, for which the largest eigenvalue is $d$.

I'll first explain some conjectures on the extremal eigenvalue distributions of adjacency matrices of random $d$-regular graphs. In the second part of the talk, I will discuss a new proof of Alon's second eigenvalue conjecture, which asserts that with high probability, the second eigenvalue of a random $d$-regular graph concentrates around $2\sqrt{d-1}$. Our proof shows that the fluctuations of these extreme eigenvalues are bounded by $N^{−2/3+\varepsilon}$, where $\varepsilon>0$ can be arbitrarily small. This gives the same order of fluctuation as the eigenvalues of matrices from the Gaussian Orthogonal Ensemble. This work is based on joint research with Theo McKenzie and Horng-Tzer Yau.

个人简历：黄骄阳，2014年在麻省理工学院数学系获得学士学位，2019年在哈佛大学数学系获得博士学位，2019-2022年在普林斯顿高等研究院和纽约大学做博士后。现为宾夕法尼亚大学统计和数据科学系助理教授，入选2024Sloan研究奖。研究方向为随机矩阵理论、随机图、交互粒子系统、深度神经网络优化、后验采样和大规模逆问题的不确定性量化。