##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Mikhail Mazin

#### Institute for Mathematical Sciences, Stony Brook University

## Topology of Compactified Jacobians and Catalan Combinatorics

##### Abstract:

Compactified Jacobian of a singular rational curve is homeomorphic to the direct product of Jacobi factors of singularities of the curve. Therefore, to study the topology of Compactified Jacobians in this case, it suffices to study topology of Jacobi factors, which can be defined very explicitly as subvarieties in Grassmanians. J. Piontkowski showed that in certain cases Jacobi factors can be decomposed into affine cells enumerated by semimodules over the semigroup of the singularity. We proved that for quasihomogeneous plane curve singularities the cells can also be enumerated by Young diagrams inscribed in a right triangle, and dimensions of cells can be computed in a combinatorial way. The resulting combinatorial model is closely related to cell decompositions of Hilbert schemes of points on a complex plane, and to rational generalizations of Garsia-Haiman's q,t-Catalan numbers. In this talk, I will discuss our results on topology of Compactified Jacobians and, time permitting, mention connections to the theory of finite dimensional representations of rational Cherednik algebras and homological knot invariants. The talk is based on a joint work with Eugene Gorsky.

Hosts: Adriano Garsia and Jeff Remmel

### January 15, 2013

### 1:00 PM

### AP&M 7321

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