This seminar is hosted weekly on Fridays 12:30  2pm Wednesdays 4  5:30pm.
Contact if you would like to be added to the mailing list. It is a
hybrid model with speakers remote on Zoom but an inperson viewing in Evans 748
Evans 740.
Date  Presenter  Topic (hover for abstract)  Links 

Feb 4  Tarun Kathuria  Matrix Concentration Inequalities and Free Probability  arxiv 
Feb 11  Tarun Kathuria  Matrix Concentration Inequalities and Free Probability (part 2)  
Feb 18  Jorge Vargas 
Strong convergence of random matrices from spectral inclusion of linear
pencils
In this talk I will discuss one of the main pieces in the proof of Theorem
1.3 in [BBvH21], which is a statement about strong convergence of a general
family \(H_1^N,\dots, H_k^N\) of Gaussian random matrices. In particular,
this result implies that random matrices obtained by evaluating a
noncommutative polynomial in the \(H_i^N\) do not have eigenvalue outliers,
and gives some machinery to compute the edge of the bulk. This talk will be
selfcontained and its key ideas, which can be applied to other random
matrix models, go back to a paper of Haagerup and Thorbjornsen from 2005.
More specifically, the starting point of the talk will be a bound on the
norm of the resolvant of linear pencils with matrix coefficients on the
\(H_i^N\) which will be used as a blackbox (this bound is proven in
[BBvH21] using the interpolation technique that Tarun explained in the
previous two talks), from where I will explain how to control quantities of
the form \(\p(H_1^N, \dots, H_k^N)\\), where \(p\) is a noncommutative
polynomial.

arxiv 
Feb 25  Sidhanth Mohanty 
Noise sensitivity of the top eigenvector of a random matrix
Suppose \(M\) is a random matrix and \(M'\) is a matrix we get after
applying some noise, when is the top eigenvector of \(M'\) barely different
from that of \(M\)? When is it completely uncorrelated? The behavior
observed is a phase transition from nearperfect correlation to nearzero
correlation as we increase the amount of noise, and the location of this
phase transition is now understood for several random matrix models of
interest. Notably, when \(M\) is a Wigner matrix, and \(M'\) is obtained by
resampling \(k\) random entries of \(M\), the top eigenvectors of \(M\) and
\(M'\) are closely aligned when \(k \ll n^{5/3o(1)}\) and are
nearorthogonal when \(k \gg n^{5/3}\). The goal of this talk is to go over
a proof of this fact when M is a Gaussian matrix.

book arxiv 1 arxiv 2 
March 4  Theo McKenzie 
Universality of nodal count distribution in large metric graphs
We examine the nodal count, namely the number of 0's, of eigenfunctions of
quantum graphs. Alon, Band, and Berkolaiko conjecture that the normalized
distribution of nodal counts for the first \(n\) eigenfunctions converges to
the Gaussian distribution as n goes to infinity. We give their proof of this
conjecture for certain limiting cases, and also give a method for computing
these distributions efficiently.

arxiv 
March 11  Theo McKenzie 
Universality of nodal count distribution in large metric graph (part 2)
In the second part of my talk, we will prove that the nodal surplus is equal
to the number of positive eigenvalues of the Hessian on the space of
equipartitions. In order to define an equipartition, we examine properties
of the zero set of eigenfunctions on quantum graphs. Finally, we will show
how this Hessian can be calculated in the torus of phases introduced in the
first part of the talk.


March 18  Elizabeth Yang 
Quantum Tanner Codes
To start, I'll first spend some time introducing the CSS code framework for
quantum coding, and then describe the new "leftright Cayley complex"
construction and how to obtain the rate and distance bounds for its
corresponding quantum code. This new construction also has connections to
the recent breakthrough of Dinur et al, in which they construct the first
known \(c^3\) locally testable codes. Similar ideas can be used both to
prove linearly growing minimum distance for the quantum code, and to prove
local testability for Dinur et al's classical code.

arxiv (no video) 
March 25  Spring Break!  
April 1  Elizabeth Yang 
Quantum Tanner Codes (part 2)
This week, we'll dive into the actual construction of quantum Tanner codes
by Leverrier and Zemor, based on the square Cayley complex of Dinur et al.
The main technical challenge we will tackle is understanding the quantum
distance of these quantum Tanner codes.


April 6  Omar Alrabiah 
Quantum Tanner Codes (part 3)
In this talk, we are going to wrap up the series on the Quantum Tanner Codes
of (Leverrier and Zemor, 2022) by showing their quantum distance lower
bound. First, we will prove the robustness proposition stated in the last
talk. After that, we are going to show the distance lower bound for the
Expander Codes of (Sipser and Spielman, 1996) as a warmup toward the
quantum distance lower bound. Finally, we will end by proving the quantum
distance lower bound.


April 13  Omar Alrabiah 
Quantum Tanner Codes (part 4)
Final part in this series, see part 3 for abstract.


April 20  Rikhav Shah 
Worst case to average case reduction for matrix multiplication
Say one is given a defective oracle for multiplying matrices over some
finite field \(\mathbb{F}\). Specifically, say that it only outputs the
correct value of \(AB\) for just 1% of possible pairs of matrices \(A,B\).
Can the oracle be repaired? The answer is yes: the oracle can be efficiently
turned into a randomized algorithm which computes \(AB\) in all cases with
probability 99%.

arxiv 
April 27  Zach Stier 
Short paths in LPS graphs and \(PU(2)\)
Carvalho Pinto and Petit recently devised a method for efficiently computing
paths of length \(\frac{7}{3}\log_pq\) in the LPS Ramanujan graph
\(X^{p,q}\). This approach to the problem in a discrete setting has
adaptations to continuous settings such as quantum hardware design, where
the path length grows as \(\frac{7}{3}\log_{59}\frac{1}{\varepsilon^3}\) for
precision $\varepsilon$. In this talk we will cover these algorithms for
factorizations in LPS graphs and \(PU(2)\) (singlequbit gates, via golden
gates), and time permitting discuss additional numbertheoretic aspects of
these constructions.

paper 1 paper 2 arxiv 