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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.