##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Jiarui Fei

#### UC Riverside

## Vanishing cycles and cluster algebras

##### Abstract:

For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. We use Ringel-Hall algebras as the main organizing tools. The wall-crossing formula leads to a categorification of quantum cluster algebras under the assumption of existence of certain potential. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a formula relating the representation Grassmannians under sink-source reflections. In this talk, I will start with basic definitions of mutations of quivers with potentials, vanishing cycles, and Hall algebras.

Host: Dan Rogalski

### November 18, 2013

### 2:00 PM

### AP&M 7218

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