Using AI to study topological 4-manifolds | Slava Krushkal, University of Virginia | IAS/PCMI

Title: Using AI to study topological 4-manifolds Presented at IAS-PCMI by Slava Krushkal, University of Virginia Abstract: The Round Handle Problem asks whether a given link in the 3-sphere is slice in the 4-manifold obtained by attaching certain round handles to the 4-ball. The problem is significant because of its implications for the 4-dimensional topological surgery and s-cobordism conjectures for free groups. I will discuss an algebraic-combinatorial formulation of the problem, ongoing AI-assisted work on it, and the results obtained to date. – PCMI 2026 Research Topic: Knotted Surfaces in Four-Manifolds Organizers: R. İnanç Baykur, University of Massachusetts Amherst; Kyle Hayden, Rutgers University – Newark; András Stipsicz, Rényi Institute of Mathematics; Gordana Matic, University of Georgia; Masaki Taniguchi, Kyoto University; and Ian Zemke, University of Oregon. The 2026 Graduate portion of PCMI considers ways to construct and represent interesting surface embeddings in four-dimensinal spaces. The topological theory is addressed along with invariants that distinguish topologically equivalent smooth embeddings. Complex, and symplectic notions are covered. Techniques ranging from gauge theory, to Floer theory, to mapping class groups are considered as well. The core of the program is nine graduate mini-courses taught by a diverse group of leading researchers in the field renowned for their clear and engaging lecturing styles. In parallel, are thematic workshops aimed at more senior researchers as well as activities for undergraduate students.