ASD Connections and Cosmetic Surgery - March 28, 2025

This lecture by Mike Miller Eismeier explores the cosmetic surgery conjecture in knot theory and 3-manifolds, focusing on advanced mathematical techniques to address unresolved cases. Discover how the conjecture predicts that for a knot K in a 3-manifold, the oriented diffeomorphism type of surgery determines the surgery slope (up to oriented diffeomorphism). Learn about the sequence of restrictions from Heegaard Floer homology that have narrowed down potential counterexamples in the 3-sphere to specific surgery slopes. Understand how Miller Eismeier's work with quantitative enhancements of instanton homology has further ruled out cases where r = 1/n, leaving only the possibility of 2-surgery as a potential counterexample. The presentation also covers the limitations of current approaches and discusses promising directions for future research in this area of low-dimensional topology.