Georgia Institute of Technology College of Sciences

This talk is about an application of combinatorial algebraic geometry to complex/arithmetic dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic ramification point. Per_n is an affine algebraic curve, defined over Q, parametrizing degree-2 rational maps with an n-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is G_n is irreducible over Q? (2) Is Per_n connected? We show that if G_n is irreducible over Q, then Per_n is irreducible over C, and is therefore connected. In order to do this, we find a Q-rational smooth point on a projective completion of Per_n — this Q-rational smooth point represents a special degeneration of degree-2 self-maps.

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.

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.

The brain performs efficient, adaptable, and robust computations of noisy sensory information in changing environments. While artificial neural networks have achieved remarkable successes in recent years, the brain's computational capacity is yet to be matched. To understand mechanisms underlying the exquisite computational efficiency and flexibility of the brain, complex architecture and dynamics of the biological neural networks should be studied. In this talk, I will give a broad overview of recent research projects from my group, that investigate links between neural coding and network structures using data-driven modeling.

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.

The Fox trapezoidal conjecture is a longstanding open problem about the coefficients of the Alexander polynomial of alternating links. In this talk, we will discuss recent work which settled this conjecture for “special alternating links”. The first tool is a graph theoretic model of the Alexander polynomial of an alternating link discovered by Crowell in 1959. The second is the theory of Lorentzian polynomials, developed by Brändén and Huh in 2019 and a key part of Huh’s Fields medal work. We will show how a version of Crowell’s model produces a refinement of the Alexander polynomial of special alternating links that is Lorentzian, from which the result follows quickly.

Assuming the Riemann Hypothesis, Montgomery established results concerning the pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results for all level correlations of automorphic $L$-functions. We discover surfaces associated with the zeros of automorphic $L$-functions. In the case of pair correlation, the surface displays Gaussian behavior. For triple correlation, these structures exhibit characteristics of the Laplace and Chi-squared distributions, revealing an unexpected phase transition. This is joint work with Debmalya Basakand Alexandru Zaharescu.

Volume polynomials constitute a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties arising from convex bodies and projective varieties. I will introduce new entropy inequalities satisfied by volume polynomials, discuss applications to the combinatorics of algebraic matroids, introduce the new class of analytic matroids, and pose several open questions (based on joint with Lukas Grund, Mateusz Michalek, Henrik Süss, and Botong Wang).

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Depending on how long it takes to discuss these items, I will also attempt to include discussions on:

• Biran-Giroux decompositions of symplectic manifolds • Boothby-Wang bundles and contact plumbings of these • Milnor's fibration theorem for isolated singularities and connections to open book decompositions and Lefschetz fibrations • Open questions and interesting avenues of research

Most of our discussion will, as a side effect, outline the topological structure behind Type IIA String theory (the "topological A-model") which requires a 6-dimensional Calabi-Yau (Kahler) background.

Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. I will talk about a recent joint work with Anton Gorodetski and Victor Kleptsyn, where we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.