Georgia Institute of Technology College of Sciences

Algebraic dynamics is the study of iterating self-maps of algebraic varieties. Tropical geometry provides tools to study algebraic varieties via polyhedral complexes called "tropicalizations". We identify a class of algebraic self-maps that tropicalize well, and can therefore be studied via induced piece-wise-linear dynamics.  Our motivating examples are dynamical systems on the moduli spaces M_{0,n} of point-configurations on P^1. This is work-in-progress joint with Charles Favre.

Many emerging applications of artificial intelligence, from analyzing scientific literature to developing complex software, rely on language models to process increasingly large amounts of information. Yet the ability to accept a long input, or context, does not guarantee that a model can use it effectively. In this talk, I will investigate how language models use long contexts through mathematical models of transformers, the architecture underlying many language models.

I will first study how language models learn from examples supplied in their input without changing their parameters, an ability known as in-context learning. A longer context can contain more examples, but how rapidly can a model benefit from this growing body of evidence? Using mean-field analysis, we establish quantitative guarantees for learning from finitely many examples.

The benefit of additional information, however, also depends on whether a model can focus on useful evidence amid irrelevant material. I will therefore study temperature scaling, a practical strategy that adjusts how selectively a model weighs different parts of its input. Drawing on extreme-value theory, we identify a critical scaling with context length at which only several selected pieces of information remain influential.

I will conclude by connecting these results to my broader research on mathematical machine learning and highlighting how applied mathematics can inform the design of reliable learning algorithms.

This talk presents the construction of large global classical flows for the compressible Euler equations and their stability under multidimensional perturbations. The global solutions arise from supersonic radial rarefactive data, without any smallness assumption on the amplitude. These solutions serve as backgrounds for a nonlinear stability problem under general three-dimensional perturbations. Quantitative lower bounds on the classical lifespan and stability timespan tend to infinity as the perturbation size approaches zero. The analysis accommodates a point vacuum at the origin, and the rarefactive structure yields a longer stability timespan than the corresponding energy estimate for a general smooth background. 

The final part addresses the evolution beyond classical breakdown for the three-dimensional pressureless equations. The structure of newly forming singularities is characterized for an open dense set of smooth initial data. For analytic data satisfying suitable nondegeneracy conditions, a local continuation past the first singularity is constructed in which a positive amount of mass concentrates on an evolving two-dimensional surface. 

The radial existence result is joint with Geng Chen, Yanbo Hu, and Yannan Shen. The pressureless results are joint with Geng Chen and Shoujun Huang.

In 1996, Merel showed that for any elliptic curve defined over a number field $F$ of degree $d$, the size of the torsion subgroup is bounded by a constant $B(d)$ that depends only on $d$. It is conjectured that one can choose $B(d)$ to be polynomial in the degree $d$. For the special class of elliptic curves with $j$-invariant in $\mathbb{Q}$, polynomial bounds were first established by Clark and Pollock in 2018. In this talk, I will discuss recent improvements to these bounds that are essentially optimal for the exponent of the torsion subgroup.

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent

set of size $n$. 

We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute constant.  

As a consequence, we  determine the tower growth rate of $r_k(k+1,n)$, which completely solves the problem of establishing the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, first posed by Erd\H{o}s and Hajnal in 1972.