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Explore the algebraic structure of homeomorphism and diffeomorphism groups in this mathematical lecture that examines how Floer theory invariants and continuous symplectic topology methods demonstrate the failure of simplicity in the group of area-preserving homeomorphisms of the two-dimensional disc. Discover how symplectic Weyl laws enable the extension of the Calabi invariant from smooth settings to area-preserving homeomorphisms, and learn about the extension of helicity—a conserved quantity from three-dimensional Euler equations—from smooth volume-preserving flows to continuous ones. Delve into the historical context of research from the 1960s and 1970s that established the simplicity of most homeomorphism and diffeomorphism groups, while understanding how modern techniques from Floer theory provide new insights into this notable open case. Gain insight into the connections between infinite-dimensional Morse theory analogues and continuous symplectic topology methods that reveal fundamental properties of these mathematical structures.
