Monday, October 12th, 2026
Wednesday, October 14th, 2026
Math Event: Algebraic Geometry Seminar: Dmitry Kaledin - Towards non-commutative Hodge theory
Time: 4:00 PM - 5:00 PM
Location:
Speaker: Dmitry Kaledin
Abstract: In non-commutative algebraic geometry, an algebraic variety is replaced by a DG algebra A, and the role of de Rham cohomology is played by periodic cyclic homology HP(A). It is a standard expectation that all the additional motivic-type structure on de Rham cohomology should also exist in non-commutative generality. There has been a lot of progress on this in the p-adic situation, but the archimedian case is much more problematic. The missing ingredient is complex conjugation. While a Hodge filtration on periodic cyclic homology exists by definition, we also expect that HP(A) for a good A over C carries a functorial R-structure, and it is not clear at all where it might come from. I am going to present a proposal towards this based on L^2-completed version of HP. This is still work-in-progress, but the proposal looks viable, so hopefully, this won't be a complete waste of audience's time.
Speaker: Dmitry Kaledin
Abstract: In non-commutative algebraic geometry, an algebraic variety is replaced by a DG algebra A, and the role of de Rham cohomology is played by periodic cyclic homology HP(A). It is a standard expectation that all the additional motivic-type structure on de Rham cohomology should also exist in non-commutative generality. There has been a lot of progress on this in the p-adic situation, but the archimedian case is much more problematic. The missing ingredient is complex conjugation. While a Hodge filtration on periodic cyclic homology exists by definition, we also expect that HP(A) for a good A over C carries a functorial R-structure, and it is not clear at all where it might come from. I am going to present a proposal towards this based on L^2-completed version of HP. This is still work-in-progress, but the proposal looks viable, so hopefully, this won't be a complete waste of audience's time.