Algebra/Topology Seminar

Speaker: Boris Tsygan

Title: Gauss-Manin connection in non-commutative geometry

Abstract: The Gauss-Manin connection on the De Rham cohomology of a family of varieties was defined by Manin in 1957 and interpreted by Grothendieck and Katz-Ono in terms of Cartan calculus of differential forms and vector fields. When one replaces a variety by an associative algebra, the De Rham cohomology gets replaced by periodic cyclic homology. The Gauss-Manin connection in this context was defined by Getzler in 1991. The subject had been revisited several times, Cartan calculus in noncommutative setting being subtle and involved. I will review the current state of affairs, including some intriguing similarities between explicit formulas for the connection and WKB asymptotic expansions, relation to noncommutative crystalline cohomology and (time permitting) to noncommutative Hodge theory.