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This is an announcement of Wenjun Cai’s Dissertation Defense.
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<div class="">Candidate: Wenjun Cai</div>
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<div class="">Date: Tuesday, January 24, 2023</div>
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<div class="">Time: 3 pm CST</div>
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<div class="">Location: JCL 298</div>
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<div class="">Title: On the spectrum of singular values of multi-z-shaped graph matrices</div>
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<div class="">Abstract: </div>
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<li class=""><font face="arial, sans-serif" class="">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.</font>
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<p class=""><font face="arial, sans-serif" class="">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.</font></p>
<p class=""><font face="arial, sans-serif" class="">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 ◦<span style="vertical-align: -3pt;" class="">R </span>operation,
which mixes two distributions Ω and Ω<span style="vertical-align: 4pt;" class="">′ </span>via a random orthogonal matrix.</font></p>
<p class=""><font face="arial, sans-serif" class="">This ◦<span style="vertical-align: -3pt;" class="">R </span>operation is closely connected to free probability theory, where ◦<span style="vertical-align: -3pt;" class="">R </span>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. </font></p>
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<div class="">Advisors: Aaron Potechin and Alexander Razborov</div>
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<div class="">Committee Members: Aaron Potechin, Alexander Razborov, and Laszlo Babai</div>
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