Georgia Institute of Technology College of Sciences

The study of contact and symplectic manifolds has relied heavily on understanding Legendrian and Lagrangian submanifolds in them -- both for constructing the manifolds using these submanifolds, and for computing invariants of the ambient space in terms of invariants of the submanifolds. This thesis explores the construction of Legendrian submanifolds in high dimensional contact manifolds (greater than 3) in two directions. In one, using open book decompositions, we generalise a doubling construction defined by Ekholm and show that the Legendrians obtained are trivial. In the second, which is joint work in progress with Hughes, we use the doubling and twist spun constructions to build a large family of Legendrians, compute their sheaf-theoretic invariants to distinguish them using techniques of Casals-Zaslow, and study their exact Lagrangian fillability properties.

Zoom link:

https://gatech.zoom.us/j/93109756512?pwd=Skljb0tVdjZVNEUvSm9tNnFHZFREUT09 

Suppose you have a set S of integers from {1,2,...,N} that contains at least N / C elements. Then for large enough N, must S contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?

In 1953, Roth showed this is the case when C is roughly (log log N). Behrend in 1946 showed that C can be at most exp(sqrt(log N)). Since then, the problem has been a cornerstone of the area of additive combinatorics. Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on C to C = (log N)^(1+c) for some constant c > 0.

This talk will describe a new work showing that C can be as big as exp((log N)^0.08), thus getting closer to Behrend's construction. Based on joint work with Zander Kelley

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

►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:
· Michael Lacey (advisor)
· Chris Heil
· Ben Krause
· Doron Lubinsky
· Shahaf Nitzan