[Colloquium] Reminder - Wenjun Cai Dissertation Defense/Jan 24, 2023

[Megan Woodward]

This is an announcement of Wenjun Cai’s Dissertation Defense.
===================================================
Candidate: Wenjun Cai

Date: Tuesday, January 24, 2023

Time: 3 pm CST

Location: JCL 298

Title: On the spectrum of singular values of multi-z-shaped graph matrices

Abstract:

  *   This thesis studies graph matrices. Graph matrices are a type of random matrices that were invented as a powerful tool for analyzing large and complicated moment matrices which often arise in the analysis of the Sum of Square Hierarchy. They are also useful for other methods involving higher moments. Previous studies on graph matrices mainly focused on their norm bounds. In this thesis, we further investigate their spectrum.

We start with determining the spectrum of singular values of Z-shaped and multi-Z- shaped graph matrices with input distribution being ±1 as its dimension parameter n → ∞. This result can be seen as an analog of Wigner’s Semicircle Law in the special case of ±1 distribution, instead of any arbitrary distribution with mean 0 and variance 1.

We then generalize our result to multi-Z-shaped graph matrices with arbitrary input distributions with variance 1 and 0 odd moments. We achieve this using the ◦R operation, which mixes two distributions Ω and Ω′ via a random orthogonal matrix.

This ◦R operation is closely connected to free probability theory, where ◦R corresponds to the free multiplicative convolution. As a part of our analysis, we prove a new direct formula for the moment concerning two freely independent variables. We also prove some new results on non-crossing partitions which is an essential part of free probability theory.


Advisors: Aaron Potechin and Alexander Razborov

Committee Members: Aaron Potechin, Alexander Razborov, and Laszlo Babai
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://mailman.cs.uchicago.edu/pipermail/colloquium/attachments/20230124/6b17ff31/attachment-0001.html>