An arrangement of hypersurfaces in projective space is simple normal crossing (rough idea: hypersurfaces are generic so meet transversally) if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

(joint with T. Kahle, B. Sturmfels, M. Wiesmann; Foundations of Computational Math, to appear)

The quantum zipper is a remarkable gluing principle in two-dimensional random geometry. Roughly speaking, it says that two independent random surfaces, called Liouville quantum gravity (LQG) surfaces, can be conformally welded along their boundaries, and that the resulting interface is described by a Schramm–Loewner evolution (SLE) curve. Conversely, cutting an LQG surface along an appropriate SLE curve produces two independent LQG surfaces.

I will mostly give a friendly introduction to this correspondence, focusing on the underlying geometric ideas rather than technical details. I will then briefly discuss ongoing work aimed at extending the quantum zipper beyond the regime that is currently understood. I will conclude with some speculative questions about possible higher-dimensional analogues, particularly in dimension four, where genuinely new topological input appears to be necessary.

Despite rapid advances in geometric deep learning, performing statistical analysis on geometric data—where observations consist of curves, surfaces, or shape graphs—remains fundamentally challenging. This difficulty stems from the non-Euclidean structure of shape spaces, which are defined as equivalence classes under invariance groups, most notably the infinite-dimensional group of reparameterizations. We will discuss addressing these challenges with tools from geometric measure theory. By embedding shapes into measure spaces, we may obtain a robust representation for deep learning tasks which are inherently invariant to reparameterization.

Despite the tremendous success of artificial intelligence (AI) in science, engineering, and technology in the past decade, its explainability, generalizability, and reliability have been a major concern. The solution to these challenges holds the future of AI. Topological deep learning (TDL), a new paradigm in rational learning introduced by us in 2017, offers interpretable and generalized AI approaches. TDL utilizes topological data analysis (TDA), which is originally rooted in persistent homology, an algebraic topology technique. However, persistent homology has many limitations, including the lack of localization, being restricted to point cloud data, and the inability to represent non-topological information. To address these challenges, we generalized TDA to combinatorial spectral theory (e.g. Topological Laplacian and Dirac), differential topology (e.g. de Rham-Hodge theory), and geometric topology (e.g. Khovanov homology) to handle data on graphs, differentiable manifolds, and curves embedded in 3-space, respectively (see Artif Intell Rev 59, 58, (2026) for a review). To further advance AI through modern mathematical theories, we introduced commutative algebra as a new frontier in data science and machine learning.

Guowei Wei received his Ph.D. from the University of British Columbia and is currently a Professor and GRA Eminent Scholar at the University of Georgia. Dr. Wei is a Fellow of SIAM and AIBME. He has mentored more than 170 research students, postdoctoral researchers, and visiting scholars. His research focuses on the mathematical foundations of bioscience and artificial intelligence (AI). Dr. Wei and his collaborators have pioneered several mathematical AI paradigms, including topological deep learning (TDL), manifold topological learning (MTL), and commutative algebra learning (CAL), which integrate modern mathematical structures with AI to address challenging problems involving real-world data. Their mathematical AI approaches have led to victories in the D3R Grand Challenges, a worldwide annual competition series in computer-aided drug design. By combining TDL, genotyping, and computational biophysics, the Wei team elucidated the mechanisms underlying SARS-CoV-2 evolution and successfully predicted emerging dominant SARS-CoV-2 variants months in advance.

Refreshments are served at 3:00 PM at the 3rd floor lounge.

Refreshments are served at 3:00 PM. Joint Department Colloquium and Topology and Geometry seminar.

Online session; the seminar site does not yet provide a Zoom link.