Georgia Institute of Technology College of Sciences

The representations of quantum groups are important in topology, namely, they can be used to construct quantum invariants of links. This relationship goes both ways: for example, the equivariant tensor category of representations of $U_q(\mathfrak{sl}_2)$ can be understood as a category of tangles. We will discuss a landmark result by Kuperberg who constructed graphical calculuses which describe the representation theory of the rank-2 simple Lie algebras.

The use of neural networks for solving partial differential equations (PDEs) has attracted considerable attention in recent years. In this talk, I will first highlight their advantages over traditional numerical methods, including improved approximation rates and the potential to overcome the curse of dimensionality. I will then discuss the challenges that arise when applying neural networks to PDEs, particularly in training. Because training is inherently a highly nonconvex optimization problem, it can lead to poor local minima with large training errors, especially in complex PDE settings. To address these issues, I will demonstrate how incorporating mathematical insight into the design of training algorithms and network architectures can lead to significant improvements in both accuracy and robustness.

In recent years, the theories of Lorentzian polynomials and combinatorial Hodge theory have been developed and utilized to resolve long-standing conjectures in matroid theory, related to log-concavity inequalities and sampling algorithms. The overarching idea in these theories is to extract the conjectured results from basic eigenvalue bounds on certain natural matrices associated to matroids. Since then, Lorentzian polynomials have been generalized beyond matroids to simplicial complexes of various types, implying old and new results on various combinatorial structures such as linear extensions of posets. That said, many questions remain open. In this talk, we will describe this generalized theory and discuss how it can be used to prove various combinatorial results. No knowledge of matroid theory will be assumed. Joint work with Kasper Lindberg and Shayan Oveis Gharan, and also with Petter Brändén.

Transformers serve as the foundational architecture for large language and video generation models, such as GPT, BERT, SORA, and their successors. While empirical studies have shown that real-world data and learning tasks exhibit low-dimensional geometric structures, the theoretical understanding of transformers in leveraging these structures remains largely unexplored. In this talk, we present a theoretical foundation for transformers in two key scenarios: (1) regression tasks with noisy input data lying near a low-dimensional manifold, and (2) in-context learning (ICL) for regression of Hölder functions on manifolds. For the first setting, we prove that approximation and generalization bound that depend crucially on the intrinsic dimension of the manifold, demonstrating that transformers can effectively learn from data perturbed by high-dimensional noise. For the second setting, we derive generalization error bounds for ICL in terms of prompt length and the number of training tasks, revealing that transformers achieve the minimax optimal rate for Hölder regression—scaling exponentially with the intrinsic rather than ambient dimension. Together, these results provide foundational insights into how transformers exploit low-dimensional geometric structures in learning tasks, advancing our theoretical understanding of their remarkable empirical success.