## Topology & Geometric Group Theory Seminar

## Fall 2010

### 1:30 – 2:30, Malott 253

Tuesday, August 31

**Emily
Dryden**, Bucknell University

*Upper bounds for eigenvalues of submanifolds*

The problem of finding upper bounds for eigenvalues of the Laplace
operator has a rich history. In 1970, Joseph Hersch showed that for
the sphere, the product of its surface area and lowest nonzero
eigenvalue is at most 8π, regardless of the metric. Moreover, the
bound is realized only by the usual constant curvature metric. In
contrast, Bruno Colbois and Józef Dodziuk showed that for
manifolds of dimension three and higher, the eigenvalues can be
unbounded unless additional geometric constraints are imposed. We
discuss upper bounds on eigenvalues in the setting of compact
submanifolds of Euclidean space. No previous experience with
eigenvalues on manifolds will be assumed.

This is joint work with Bruno Colbois and Ahmad El Soufi.

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