## Seminars and Colloquia by Series

### Matroids, Matrices, and Partial Hyperstructures

Series
Dissertation Defense
Time
Wednesday, July 5, 2023 - 02:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Tianyi ZhangGeorgia Tech

I will talk about the application of algebra and algebraic geometry to matroid theory. Baker and Bowler developed the notions of weak and strong matroids over tracts. Later, Baker and Lorscheid developed the notion of foundation of a matroid, which characterize the representability of the matroid. I will introduce a variety of topics under this theme. First, I will talk about a condition which is sufficient to guarantee that the notions of strong and weak matroids coincide. Next, I will describe a software program that computes all representations of matroids over a field, based on the theory of foundations. Finally, I will define a notion of rank for matrices over tracts in order to get uniform proofs of various results about ranks of matrices over fields.

### Set Images and Convexity Properties of Convolutions for Sum Sets and Difference Sets

Series
Dissertation Defense
Time
Friday, June 23, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chi-Nuo LeeGeorgia Tech

Many recent breakthroughs in additive combinatorics, such as results relating to Roth’s theorem or inverse sum set theorems, utilize a combination of Fourier analytical and physical methods. Physical methods refer to results relating to the physical space, such as almost-periodicity results regarding convolutions. This thesis focuses on the properties of convolutions.

Given a group G and sets A ⊆ G, we study the properties of the convolution for sum sets and difference sets, 1A ∗1A and 1A ∗1−A. Given x ∈ Gn, we study the set image of its sum set and difference set. We break down the study of set images into two cases, when x is independent, and when x is an arithmetic progression. In both cases, we provide some convexity result for the set image of both the sum set and difference set. For the case of the arithmetic progression, we prove convexity by first showing a recurrence relation for the distribution of the convolution.

Finally, we prove a smoothness property regarding 4-fold convolutions 1A ∗1A ∗1A ∗1A. We then construct different examples to better understand possible bounds for the smoothness property in the case of 2-fold convolutions 1A ∗ 1A.

Committee

Prof. Michael Lacey

Prof. Josephine Yu

Prof. Anton Leykin

Prof. Will Perkins

### Functional Ito Calculus for Lévy Processes (with a View Towards Mathematical Finance)

Series
Dissertation Defense
Time
Thursday, June 22, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006/Zoom
Speaker
Jorge Aurelio Víquez BolañosGeorgia Tech

Zoom link.  Meeting ID: 914 2801 6313, Passcode: 501018

We examine the relationship between Dupire's functional derivative and a variant of the functional derivative developed by Kim for analyzing functionals in systems with delay. Our findings demonstrate that if Dupire's space derivatives exist, differentiability in any continuous functional direction implies differentiability in any other direction, including the constant one. Additionally, we establish that co-invariant differentiable functionals can lead to a functional Ito formula in the Cont and Fournié path-wise setting under the right regularity conditions.

Next, our attention turns to functional extensions of the Meyer-Tanaka formula and the efforts made to characterize the zero-energy term for integral representations of functionals of semimartingales. Using Eisenbaum's idea for reversible semimartingales, we obtain an optimal integration formula for Lévy processes, which avoids imposing additional regularity requirements on the functional's space derivative, and extends other approaches using the stationary and martingale properties of Lévy processes.

Finally, we address the topic of integral representations for the delta of a path-dependent pay-off, which generalizes Benth, Di Nunno, and Khedher's framework for the approximation of functionals of jump-diffusions to cases where they may be driven by a process satisfying a path-dependent differential equation. Our results extend Jazaerli and Saporito's formula for the delta of functionals to the jump-diffusion case. We propose an adjoint formula for the horizontal derivative, hoping to obtain more tractable formulas for the Delta of strongly path-dependent functionals.

Committee

• Prof. Christian Houdré - School of Mathematics, Georgia Tech (advisor)
• Prof. Michael Damron - School of Mathematics, Georgia Tech
• Prof. Rachel Kuske - School of Mathematics, Georgia Tech
• Prof. Andrzej Święch - School of Mathematics, Georgia Tech
• Prof. José Figueroa-López - Department of Mathematics and Statistics, Washington University in St. Louis
• Prof. Bruno Dupire - Department of Mathematics, New York University

### Divisors and multiplicities under tropical and signed shadows

Series
Dissertation Defense
Time
Tuesday, June 20, 2023 - 09:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 006 / Zoom
Speaker
Trevor GunnGeorgia Tech

Zoom link (Meeting ID: 941 5991 7033, Passcode: 328576)

I will present two projects related to tropical divisors and multiplicities. First, my work with Philipp Jell on fully-faithful tropicalizations in 3-dimensions. Second, my work with Andreas Gross on algebraic and combinatorial multiplicities for multivariate polynomials over the tropical and sign hyperfields.

The first part is about using piecewise linear functions to describe tropical curves in 3 dimensions and how the changes in those slopes (a divisor) lift to non-Archimedean curves. These divisors give an embedding of a curve in a 3-dimensional toric variety whose tropicalization is isometric to the so-called extended skeleton of the curve.

In part two, I describe how Baker and Lorscheid's theory of multiplicities over hyperfields can be extended to multivariate polynomials. One key result is a new proof/view of the work of Itenburg and Roy who used patchworking to construct some lower bounds on the number of positive roots of a system of polynomials given a particular sign arrangement. Another result is a collection of upper bounds for the same problem.

Committee:

• Josephine Yu
• Oliver Lorscheid
• Anton Leykin
• Greg Blekherman

### Improving and maximal inequalities in discrete harmonic analysis

Series
Dissertation Defense
Time
Wednesday, June 7, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 &amp; online
Speaker
Christina GiannitsiGeorgia Tech

►Presentation will be in hybrid format. Zoom link: https://gatech.zoom.us/j/99128737217?pwd=dllnNE1kSW1DZURrY1UycGxrazJtQT09

►Abstract: We study various averaging operators of discrete functions, inspired by number theory, in order to show they satisfy  $\ell^p$ improving and maximal bounds. The maximal bounds are obtained via sparse domination results for $p \in (1,2)$, which imply boundedness on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

We start by looking at averages along the integers weighted by the divisor function $d(n)$, and obtain a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$. We also show that the associated maximal function satisfies $(p,p)$ sparse bounds for $p \in (1,2)$. We move on to study averages along primes in arithmetic progressions, and establish improving and maximal inequalities for these averages, that are uniform in the choice of progression. The uniformity over progressions imposes several novel elements on our approach. Lastly, we generalize our setting in the context of number fields, by considering averages over the Gaussian primes.

Finally, we explore the connections of our work to number theory:   Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega$, so that every odd integer $n$ with $N(n)>N_\omega$ is a sum of three Gaussian primes with arguments in $\omega$.  This is the weak Goldbach conjecture. A density version of the strong Goldbach conjecture is proved, as well.

►Members of the committee:
· Chris Heil
· Ben Krause
· Doron Lubinsky
· Shahaf Nitzan

### Symmetric nonnegative polynomials and sums of squares: mean roads to infinity

Series
Dissertation Defense
Time
Wednesday, May 24, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Jose AcevedoGeorgia Tech
We study the limits of cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. Using some elements of the representation theory of the symmetric group we introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, which simultaneously allows us to tropicalize its dual cone. Using tropical convexity to describe the tropicalization of the dual cone to symmetric nonnegative forms we then compare both tropicalizations, which turn out to be convex polyhedral cones. We then show that the cones are different for all degrees larger than 4. For even symmetric forms we show that the cones agree up to degree $8$, and are different starting at degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares at the limit.

### Some Global Relaxation Methods for Quadratic and Semidefinite Programming

Series
Dissertation Defense
Time
Tuesday, May 9, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005 and ONLINE
Speaker
Shengding SunGeorgia Tech

Quadratic programming and semidefinite programming are vital tools in discrete and continuous optimization, with broad applications. A major challenge is to develop methodologies and algorithms to solve instances with special structures. For this purpose, we study some global relaxation techniques to quadratic and semidefinite programming, and prove theoretical properties about their qualities. In the first half we study the negative eigenvalues of $k$-locally positive semidefinite matrices, which are closely related to the sparse relaxation of semidefinite programming. In the second half we study aggregations of quadratic inequalities, a tool that can be leveraged to obtain tighter relaxation to quadratic programming than the standard Shor relaxation. In particular, our results on finiteness of aggregations can potentially lead to efficient algorithms for certain classes of quadratic programming instances with two constraints.

### Frames via Unilateral Iterations of Bounded Operators

Series
Dissertation Defense
Time
Thursday, April 27, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Victor BaileyGeorgia Tech

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, $\{f_k\}_{k \in I} \subset H$, be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, $\{T^n \varphi\}_{n=0}^{\infty} \subset H$, to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame. This is joint work with Carlos Cabrelli.

Join Zoom meeting:  https://gatech.zoom.us/j/96113517745

### Two graph classes with bounded chromatic number

Series
Dissertation Defense
Time
Monday, April 17, 2023 - 09:30 for 1 hour (actually 50 minutes)
Location
Skiles 114 (or Zoom)
Speaker
Joshua SchroederGeorgia Tech

A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.

### Counting Hamiltonian cycles in planar triangulations

Series
Dissertation Defense
Time
Thursday, April 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaonan LiuGeorgia Tech

Whitney showed that every planar triangulation without separating triangles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $n/ \log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles.