Louis H Kauffman

August 28, 2026

Title: Introduction to Khovanov Homology

Abstract: This talk is an introduction to Khovanov Homology, a knot and link invariant that arises from the Kauffman Bracket model of the Jones polynomial by categorification: The states in the bracket summation are upleveled to become objects in a category and resmoothngs of states become morphisms in the category. Khovanov defined a natural chain complex structure associated with this category, and the homology of this chain complex becomes the new invariant. The graded Euler characteristic of the Khovanov homology yields the bracket polynomial or the Jones polynomial, depending on grading choices. We will further discuss how the Khovanov homology allows the definition of the Rasmussen invariant that can estimate the minimal genus of a knot or link in the four ball. This talk will discuss Khovanov homology and Rasmussen invariant for both classical and virtual knot theory and we will prepare the ground for a following talk where the Rasmussen invariant will be used to determine the minimal number of reconnections needed to unknot certain knotted vortices.

Keywords: Jones polynomial, Kauffman Bracket polynomial, Khovanov Category, Khovanov complex, Frobenius algebra, Lee algebra, Rasmussen invariant, classical knot theory, virtual knot theory, virtual Khovanov homology, knotted vortex, reconnection number.

Kelin Xia

September 4, 2026

Title: Mathematical AI for Molecular Sciences: from topological data analysis to topological deep learning

Abstract: A central challenge in artificial intelligence (AI)-driven molecular science lies in efficiently representing molecular data and developing learning architectures that capture intrinsic structure-function relationships. In this work, we introduce advanced mathematics-based molecular representations and learning frameworks. Molecular structures and interactions are encoded using high-order topological and algebraic representations, including Rips complexes, Alpha complexes, Neighborhood complexes, Dowker complexes, Hom-complexes, Tor-algebras, Rhomboid tilings, Sheaves, etc. Building on these foundations, we design physics-informed geometric and topological deep learning models that systematically integrate high-order, multiscale, and periodic information of molecular systems. These models have been successfully applied to diverse molecular datasets across chemistry, biology, and materials science, demonstrating their versatility and effectiveness in uncovering complex structural-functional relationships.

Keywords: Topological/geometric data analysis, molecular representation, Geometric/topological deep learning.

Tancredi Schettini Gherardini

September 18, 2026

Title: Geometry-Aware Neural Networks for Geometric Analysis

Abstract: Physics-informed neural networks (PINNs) provide a mesh-free way of searching for solutions of differential equations by turning their residuals, constraints, or variational principles into an optimisation problem. While standard applications focus on Euclidean domains, many fundamental problems in geometric analysis involve nonlinear PDEs posed on manifolds having non-trivial topology and/or geometry. In this talk, after an introduction to PINNs, I will present a PINN-based approach to the problem of constructing minimal surfaces in hyperbolic space with prescribed asymptotic boundary, motivated by a conjecture of Joel Fine. This approach, developed in collaboration with Marco Usula, produced substantial nontrivial evidence corroborating the conjecture, by showing that the number of double points on the numerical surfaces matches theoretical predictions (see arXiv:2605.26234). Along the way, I will emphasise all aspects of our method that can be generalised into useful guidelines for the use of PINNs in geometric analysis and differential geometry. Numerical results like the ones mentioned above represent valuable insights into deep geometric conjectures and highlight the role of PINNs in the broader and rapidly evolving field of AI for mathematical discovery.

Keywords: PINNs, Plateau problem, neural networks, geometric analysis.