Georgia Institute of Technology College of Sciences

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

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.