Georgia Institute of Technology College of Sciences

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. 

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.

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.