Georgia Institute of Technology College of Sciences

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:

• Matt Baker (Advisor)
• Josephine Yu
• Oliver Lorscheid
• Anton Leykin
• Greg Blekherman

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

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. Ernie Croot, Advisor

Prof. Michael Lacey

Prof. Josephine Yu

Prof. Anton Leykin

Prof. Will Perkins