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Richard Rimanyi, UNC Chapel Hill – Geometric Methods in Representation Theory Seminar

August 31, 2018 @ 4:00 pm - 5:00 pm

Geometric Methods in Representation Theory Seminar

367 Phillips Hall, 4:00 pm – 5:00 pm

Speaker: Richard Rimanyi, UNC Chapel Hill Mathematics Department

Title: Motivic characteristic classes, Hall algebras, and DT type identities

Abstract:  The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are fine characteristic classes of singular varieties in cohomology and K theory—and their theory overlaps with the theory of Okounkov’s stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_\beta$, $C^o_\beta$ parameterized by positive roots $\beta$. We prove an identity in a deformed version of Kontsevich-Soibelman’s Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of $c^o_\beta$, $C^o_\beta$ classes formally depending on a stability function Z, does not depend on Z. This identity—which encodes infinitely many identities among rational functions in infinitely many variables—has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the $c^o_\beta$, $C^o_\beta$ classes as certain commutators in the CoHA, KHA.


August 31, 2018
4:00 pm - 5:00 pm
Event Category: