Final ACO Doctoral Examination and Defense of Dissertation
Title: Large-Scale Phenomena in Geometry, Probability and Combinatorics
Manuel Fernandez V
ACO PhD student, School of Mathematics
Date: April 11th, 2025
Time: 12:30pm
Location: Molecular Sciences & Engr-167 - 1224 Classroom MoSE
Zoom link: https://gatech.zoom.us/j/92358990548?pwd=gFPmLgCOCLJbFbaWWa99ABU1HwbRfA.1
Advisor: Dr. Galyna V Livshyts, School of Mathematics, Georgia Institute of Technology
Committee:
Dr. Galyna V Livshyts, School of Mathematics, Georgia Institute of Technology (advisor)
Dr. Santosh S Vempala, School of Computer Science, Georgia Institute of Technology
Dr. Benjamin McKenna, School of Mathematics, Georgia Institute of Technology
Dr. Vladimir Koltchinskii, School of Mathematics, Georgia Institute of Technology
Dr. Konstantin Tikhomirov, Department of Mathematics, Carnegie Mellon University
Reader:
Dr. Konstantin Tikhomirov, Department of Mathematics, Carnegie Mellon University
Link to thesis draft:
https://drive.google.com/file/d/1Xnd7aSd22j44FWHmrlU7Zp0j9EE0-8rH/view?u...
Abstract:
In this dissertation we study the presence of large-scale phenomena in the areas of
geometry, probability, and combinatorics. The contributions of this thesis are as follows:
In Chapter 2 we study the $\ell_0$ isoperimetric coefficient for measurable sets in $\R^n$.
Firstly, we show that the $\ell_0$ isoperimetric coefficient of an axis-aligned cube is of or
der $n^{-1/2}$, answering a question of Laddha and Vempala, and consequently improve
the best known lower bound on the coefficient for sufficiently regular convex bodies.
Secondly, we show that the $\ell_0$ isoperimetric coefficient of any measurable set is at
most of order $n^{-1/2}$. These two results imply that the cube essentially maximizes
the $\ell_0$ isoperimetric coefficient.
In Chapter 3 we consider two problems from non-asymptotic random matrix theory.
We first study the problem of estimating the distance between a random vector and
subspace in $\R^n$. We obtain a small ball estimate for the distance between a random
vector X and subspace H, where the entries of X and the vectors spanning
H are independent, inhomogeneous and heavy-tailed. Our results generalize work of
Livshyts, Tikhomirov and Vershynin by allowing H to have co-dimension up to order
n/logn in a setting where previous results required H to have co-dimension 1. Next,
we study the problem of estimating the smallest singular value of a random rectangular
matrix. We obtain a small ball estimate for the smallest singular value of a random
rectangular matrix whose entries are independent, inhomogeneous and heavy-tailed.
Our results generalize recent work of Livshyts and Livshyts, Tikhomirov and Vershynin.
In addition, our small ball estimates for rectangular matrices are the first to
match those obtained by Rudelson and Vershynin in the far more restrictive setting of
rectangular matrices with i.i.d. and sub-gaussian entries. From a larger context, the
generality of our results provides another example of the universality phenomenon in
random matrices.
In Chapter 4 we study the behavior of the clique chromatic number of a simple graph
in the binomial random graph model G_{n,p}. For ranges of sparse p we determine
the order of magnitude and asymptotics of its typical value. Our results answer a
question of Alon and Krivelevich and resolve a conjecture of Lichev, Mitsche, and
Warnke.
