**Claremont Mathematics Weekend**

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*Sponsored by Claremont Colleges*

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**January 30 - 31, 2016**

**Speaker:** **Chiu-Yen Kao****Title:** Computational Methods for Extremal Steklov Problems**Abstract:**

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the p-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the p-th Steklov eigenvalue is unique (up to dilations and rigid transformations), has p-fold symmetry, and an axis of symmetry. The p-th Steklov eigenvalue has multiplicity 2 if p is even and multiplicity 3 if p>=3 is odd.