Georgia Institute of Technology College of Sciences

In finite games, a Nash equilibrium occurs when no player can increase their payoff by changing their strategy unless others do. According to J. Nash, such a game always has at least one Nash equilibrium when mixed strategies are allowed. This talk discusses when games have an unexpected number of totally mixed Nash equilibrium points. Such games form varieties called Nash discriminants. The main goal of this talk is to discuss a vector bundle approach to exploring the geometric properties of Nash discriminants. Part of this talk is based on joint work with Irem Portakal and Luca Sodomaco. 

It has been observed that the relationship of ribbon concordance induces monotonic behavior across several link invariants. 

Resolving a famous open question of Gordon, Agol unified these observations by showing that ribbon concordance induces a partial order on knots in the 3-sphere.

Agol's argument can be extended to show that ribbon concordance likewise induces a partial order on arbitrary links in the 3-sphere.

Perhaps motivated by the slice-ribbon conjecture, it is natural and interesting to ask which knots and links are minimal with respect to this partial order. 

Leveraging recent developments in link Floer homology, I will share new examples of ribbon minimal knots and links. 

Some of this work is joint with Jaewon Lee and Alessio Di Prisa.

Large language models exhibit a remarkable ability to learn and adapt from context without updating their parameters. In this talk, I will present our recent work on both understanding how foundation models extract task information from context and designing contexts that enable continual improvement during inference. I will first discuss a geometric analysis on in-context learning, providing new insights into how task representations emerge and evolve across layers. I will then discuss how these insights motivate a broader paradigm of inference-time learning, in which context is actively constructed rather than passively consumed. Building on iterative refinement frameworks such as AlphaEvolve, we view context as an evolving memory that stores hypotheses, intermediate solutions, and feedback. Drawing inspiration from optimization and sequential Monte Carlo, we develop principled approaches for designing and updating context over time. Overall, understanding how models read context and how we can systematically write and evolve context may provide a foundation for the next generation of adaptive AI systems.

The abelian Yang-Mills-Higgs model on (1 + 2)-dimensional Minkowski space is a classical relativistic field theory, where a scalar complex field is coupled to an electromagnetic field. It admits topological soliton solutions called vortices. We present a proof of the asymptotic stability of the degree-one vortex under equivariant perturbations in the self-dual case.

(second of two talks; the first was on Sep. 22)

A random geometric graph (RGG) is generated by first sampling $n$ latent points independently and uniformly from the unit sphere in $R^d$, and then connecting each pair of points if their inner product exceeds a threshold. We study the sharp detection threshold---the largest dimension at which the RGG can be statistically distinguished from the Erdős--Rényi graph with the same edge density $p$. This threshold is conjectured to be $d \asymp (n h(p))^3$, where $h(p)$ is the binary entropy function. Previous works proved this conjecture for dense graphs with constant $p$ and, up to polylogarithmic factors, very sparse graphs with constant average degrees. In this series of two talks, I will discuss a resolution of this conjecture. This is based on joint work with Hang Du, Nike Sun, Yihong Wu, and Jiaming Xu.

A basic question in random matrix theory is how likely a matrix is to have a large rank. For many classical models, this probability decays extremely quickly, but the strongest results often rely on the entries being identically distributed. In this talk, I will discuss how to obtain the same exponential scale for matrices with fully independent, non-identically distributed entries under only a uniform anti-concentration assumption and explain the main ideas that make this possible.

We discuss a novel class of swarm-based gradient descent (SBGD) methods for nonconvex optimization. Each agent in the swarm is characterized by its position and mass. 

The dynamics combines two mechanisms: persistent transfer of mass from agents positioned on “higher ground” to those with lower objective values, and a mass-dependent time-stepping protocol. This coupling creates a dynamic distinction between “leaders” and “explorers.” Heavier agents act as leaders, use small time steps to refine promising regions near local minima, while light agents take larger steps, exploring the landscape for lower objective values. The swarm dynamics adaptively balances exploitation of local refinement with global exploration. 

We present convergence results and numerical experiments illustrating the effectiveness of SBGD for global optimization.