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Explore manifold classification problems and discrete subgroups of Lie groups through the lens of descriptive set theory in this 50-minute mathematical lecture. Delve into foundational work that establishes a framework for general study while examining Borel complexity computations for fundamental manifold classes. Learn about key computational examples including 2-manifolds up to homeomorphism, hyperbolic n-manifolds up to isometry, and manifolds with finitely generated fundamental groups up to isometry. Discover how this research strengthens existing results from Stuck–Zimmer, Andretta–Camerlo–Hjorth, Hjorth–Kechris, and Canary–Storm. Gain insight into the abundance of open questions this foundational work generates and understand the descriptive set theoretic approach to these classical geometric problems. The presentation covers joint work with Iian Smythe and demonstrates the intersection of descriptive set theory with geometric topology and Lie group theory.
