Georgia Institute of Technology College of Sciences

We will present advances on the boundedness of geometric maximal operators, focusing on a recent result from joint work with Paul Hagelstein and Alex Stokolos, which employs probabilistic techniques in the construction of Kakeya-type sets. The material presented extends ideas of M. Bateman and N. Katz.

It has long been conjectured that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping and the manifestation of long-range scattering. We will present our recent resolution of this problem, establishing that, contrary to previous expectations, solutions with sufficiently localized initial data decay polynomially in time. Time permitting, we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums.

Multimodal contrastive learning is a methodology for linking different data modalities, such as images and text. It is typically framed as the identification of a set of encoders—one for each modality—that align representations within a common latent space. In this presentation, we interpret contrastive learning as the optimization of encoders that define conditional probability distributions, for each modality conditioned on the other, in a way consistent with the available data. This probabilistic perspective suggests two natural generalizations of contrastive learning: (i) the introduction of novel probabilistic loss functions, and (ii) the use of alternative metrics for measuring alignment in the common latent space. We investigate these generalizations of the classical approach in the multivariate Gaussian setting by viewing latent space identification as a low-rank matrix approximation problem. The proposed framework is further studied through numerical experiments on multivariate Gaussians, the labeled MNIST dataset, and a data assimilation application in oceanography.

The rank-$n$ Dowling geometry $Q_n(\Gamma)$ is a matroid associated with a graph edge-labeled by elements of the finite group $\Gamma$. We determine the maximum size of an $N$-free submatroid of $Q_n(\Gamma)$ for various choices of $N$, including subgeometries $Q_m(\Gamma')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $\Gamma$ is trivial and $N=M(K_t)$, this problem reduces to Tur\'{a}n's classical result in extremal graph theory. We show that when $\Gamma$ is nontrivial, a complex dependence on $\Gamma$ emerges, even when $N=M(K_4)$.

This is joint work with Rutger Campbell and Jorn van der Pol.