##### Department of Mathematics,

University of California San Diego

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### Math 248 - Analysis Seminar

## Gautam Iyer

#### CMU

## Winding of Brownian trajectories and heat kernels on covering spaces

##### Abstract:

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a 'cute' eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the ``winding'' / ``entanglement'' of Brownian trajectories.

Andrej Zlatos

### January 9, 2018

### 8:55 AM

### AP&M 7321

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