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Explore the Gauss-Manin connection in noncommutative geometry through this mathematical seminar lecture delivered by Boris Tsygan from Northwestern University. Delve into how noncommutative geometry replaces traditional varieties with associative rings or appropriate categorical versions, while substituting De Rham cohomology with the periodic cyclic complex. Learn about Getzler's demonstration that periodic cyclic homology of algebra families carries a flat connection, analogous to how De Rham cohomology of variety families carries the Gauss-Manin connection. Examine the extensive research on periodic cyclic complex structures conducted by Dolgushev, Tamarkin, Tsygan, Kontsevich, Soibelman, Willwacher, and others, alongside recent revisitations by Tsygan and Antieu from different perspectives. Focus on explicit formulas that exhibit promising convergence properties in both p-adic and Archimedean contexts, while discovering their intriguing connections to mathematical physics constructions, D-module theory, and formal groups.
