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Explore the topology of smooth asymptotically stable vector fields on Rⁿ and their corresponding Lyapunov functions in this 27-minute conference talk. Discover how both spaces exhibit path-connectivity and simple connectivity when n ≠ 4, 5, and weak contractibility when n < 3. Learn about the mathematical proofs that combine Lyapunov theory with differential topology, drawing upon foundational work including Smale and Perelman's contributions to the generalized Poincaré conjecture, and results from Smale, Cerf, and Hatcher on diffeomorphism group topology. Examine practical applications including a partial solution to Conley's question, a parametric Hartman-Grobman theorem for nonhyperbolic asymptotically stable equilibria, and a parametric Morse lemma for degenerate minima. Gain insights into how Hatcher's work provides a solution to the celebrated Smale conjecture, demonstrating the lasting impact of Professor Smale's contributions to asymptotic stability theory and differential topology.
