Georgia Institute of Technology College of Sciences

The length spectrum of a Riemannian manifold (M,g) is the set of lengths of closed geodesics on M. A Riemannian metric said to be length spectrum rigid if the any other manifold with the same length spectrum is isometric. Our main result is that all negatively curved metrics whose length spectra satisfy an exponential separation condition are locally length spectrum rigid and therefore locally audible. The main tool is an approximate Livsic theorem with an exponential tail that improves with regularity. This is joint work with Jon DeWitt, James Marshall Reber, and Thomas O'Hare.

Given a knot diagram, there are certain questions one can ask: Is this a diagram of the unknot? What is the minimal crossing number? Is this knot a connected sum of two smaller knots? These questions are examples of problems in knot theory for which we do not have an efficient algorithm to solve. In this talk, we will see how these computationally hard problems in knot theory can serve as underpinnings for a quantum-secure cryptography system.

We will first cover the basics of quantum computing and quantum money. Then we will closely examine a quantum money scheme based on knot theory. Specifically, the protocol is based on the apparent difficulty of finding a sequence of Reidemeister/Cromwell moves going between two equivalent diagrams of a link. Finally, we will discuss problems in knot theory that are motivated by this protocol for quantum money.

We give an $\mathcal{O}_{\mathcal{H}}(n^5)$-time algorithm to test whether an $n$-vertex labeled tour graph has a full immersion isomorphic to a labeled tour graph $\mathcal{H}$. A labeled tour graph is a connected $4$-regular graph whose edges are labeled by binary vectors of fixed length. To obtain a full immersion, a vertex may be removed only by pairing up its four incident edges and "splitting off" both pairs, summing the labels within each pair. Even for one-bit labels, no polynomial-time algorithm was previously known for arbitrary fixed $\mathcal{H}$. Our results have implications for vertex-minors, whose study is motivated by applications in quantum networking.

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.