Georgia Institute of Technology College of Sciences

For a central simple algebra $A$ over a field $K$, there are two major invariants, viz., period and index. For a field $K$, the Brauer-$l$-dimension of $K$ for a prime number $l$, is the smallest natural number $d$ such that for every finite field extension $L/K$ and every central simple $L$-algebra $A$ (of period a power of $l$), we have that index($A$) divides period$(A)^d$.

If $K$ is a number field or a local field, then classical results from class field theory tell us that the Brauer-$l$-dimension of $K$ is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for $K$ a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields $K$, Parimala-Suresh have given some bounds.

Also, the u-invariant of $K$ is the maximal dimension of anisotropic quadratic forms over $K$. For example, the u-invariant of $\mathbb{C}$ is 1, for $F$ a non-real global or local field the u-invariant of $F$ is 1, 2, 4, or 8, etc.

In this talk, I will present similar bounds for the Brauer-l-dimension and the strong u-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$.

Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood. In this talk, we will explore the abundance of 2-torsion in instanton Floer homology with integral coefficients and demonstrate how this 2-torsion encodes intriguing topological information about relevant 3-manifolds and knots. This is a joint work with Fan Ye.

Thin liquid films flowing down vertical fibers spontaneously exhibit complex interfacial dynamics, leading to irregular wavy patterns and traveling liquid droplets. Such droplet dynamics are fundamental components in many engineering applications, including mass and heat exchangers for thermal desalination, as well as water vapor and particle capture. Recent experiments demonstrate that critical flow regime transitions can be triggered by varying inlet geometries and external fields. Similar interacting droplet dynamics have also been observed on hydrophobic substrates, arising from interfacial instabilities in volatile liquid films. In this talk, I will describe lubrication and weighted residual models for falling droplets. The coarsening dynamics of condensing droplets will be discussed using a lubrication model. I will also present our recent results on developing optimal boundary control and mean-field control for droplet dynamics. 

The existence of multi black hole solutions in General Relativity is one of the expectations from the final state conjecture, the analogue of soliton resolution. In this talk, I will present preliminary works in this direction via a semilinear model, the energy critical wave equation, in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

What is the minimum/maximum size of a set $A$ of integers that has the property that every integer in $\{1,2,\cdots, n\}$ can be written in at least/at most $g$ ways as a difference of elements of $A$? For the first question, we show that the limit of this minimum size divided by $\sqrt{n}$ exists and is nonzero, answering a question of Kravitz. For the second question, we give an asymptotic formula for the maximum size. We also consider the same problems but in the setting of a vector space over a finite field. During the talk we will discuss open problems and connections to coding theory and graph theory. This is joint work with Eric Schmutz.

We discuss the proof of the following Theorem

Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.

If time allows, we will also discuss how this fact is related to the study of smoothness and structural stability of attractors, along the intersection of topology and dynamics. We will be focusing on the proof and perspective of Oliva 1975, who was interested in Banach manifolds as phase-spaces of delay equations.

We will describe a recently discovered object, dual Lyapunov exponents, that has emerged as a powerful tool in the spectral analysis of  quasiperiodic operators with analytic potentials, leading to solutions of several long outstanding problems. Based on papers joint with L. Ge, J. You, and Q. Zhou

Modern Artificial Intelligence (AI) systems, such as ChatGPT, rely on artificial neural networks (ANNs), which are historically inspired by the human brain. Despite this inspiration, the similarity between ANNs and biological neural networks is largely superficial. For instance, the foundational McCulloch-Pitts-Rosenblatt unit of ANNs drastically oversimplifies the complexity of real neurons.Recognizing the intricate temporal dynamics in biological neurons and the ubiquity of feedback loops in natural networks, we suggest reimagining neurons as feedback controllers. A practical implementation of such controllers within biological systems is made feasible by the recently developed Direct Data-Driven Control (DD-DC). We find that DD-DC neuron models can explain various neurophysiological observations, affirming our theory.

Data imbalance is a fundamental challenge in data analysis, where minority classes account for a small fraction of the training data compared to majority classes. Many existing techniques attempt to compensate for the underrepresentation of minority classes, which are often critical in applications such as rare disease detection and anomaly detection. Notably, in empirical deep learning, the large model size exacerbates the issue. However, despite extensive empirical heuristics, the statistical foundations of these methods remain underdeveloped, which poses an issue to the reliability of these machine learning models.

In this talk, I will examine imbalanced classification problems in high dimensions, focusing on support vector machine (SVMs) and logistic regression. I will introduce a "truncation" phenomenon---which we verifed across single-cell tabular data, image data, and text data---where overfitting in high dimensions distorts the distribution of logits on training data. I will provide a theoretical foundation by characterizing the asymptotic distribution via a variational formulation. This analysis formalizes the intuition that overfitting disproportionately harms minority classes and reveals how margin rebalancing---a widely used deep learning heuristic---mitigates data imbalance. As a consequence, the theory offers both qualitative and quantitative insights into generalization errors and uncertainty measures such as calibration.

This talk is based on a joint work with Jingyang Lyu (3rd-year Stats PhD student) and Kangjie Zhou (Columbia Statistics): arXiv:2502.11323.

Recent advances in deep learning have introduced numerous innovative frameworks for solving high-dimensional mean-field games (MFGs). However, these methods are often limited to solving single-instance MFGs and require extensive computational time for each instance, presenting challenges for practical applications.

In this talk, I will present our recent work on a novel framework for learning the MFG solution operator. Our model takes MFG instances as input and directly outputs their solutions in a single forward pass, significantly improving computational efficiency. Our method offers two key advantages: (1) it is discretization-free, making it particularly effective for high-dimensional MFGs, and (2) it can be trained without requiring supervised labels, thereby reducing the computational burden of preparing training datasets common in existing operator learning methods. If time permits, I will also explore connections between this framework and in-context learning, highlighting its broader implications and potential for further advancements.

In 1961 Ulam proposed a mathematical simplification extracting the essential accelerating mechanism proposed by Fermi, as to explain the cause of high-energy particles in cosmic rays. In this talk, we shall describe the typical behavior of the very model introduced by Ulam in both the classical original form as well as its quantization. In the classical model, we show that typical orbits are recurrent under resonance assumptions. Meanwhile in the quantum model, the acceleration caused by resonance gets much amplified and we point out a direct relationship between the acceleration behavior of the system and the shape of its quasi-energy spectrum. This is a joint work in progress with Changguang Dong and Disheng Xu.

The 2025 Atlanta Science Festival Mathapalooza! will take place in an art gallery at the opening of an exhibition, The Art of Math.  The show will begin with magic by Matt Baker, followed by hands-on art construction, circus acts, "Math Court" skits, and 4-dimensional dance, before ending with a performance by Tracy Woodard of Bach's Crab Canon, with a discussion of its symmetry algebra.  Find out more and get tickets at https://atlantasciencefestival.org/events-2025/1113-mathapalooza-at-the-gallery/