December 12, 2013
Mathematics colloquium lecture Dec. 12
Braxton Osting, University of California, Los Angeles, will present "Extremal Eigenvalue Problems in Optics, Geometry, and Data Analysis" as part of the mathematics department Colloquium lecture series at 2:30 p.m. Thursday, Dec. 12, in 122 Cardwell Hall.
The abstract for the lecture: Since Lord Rayleigh conjectured that the disk should minimize the first eigenvalue of the Laplace-Dirichlet operator among all shapes of equal area more than a century ago, extremal eigenvalue problems have been an active research topic. In this talk, I'll demonstrate how extremal eigenvalue problems arise in a variety of contexts, including optics, geometry, and data analysis, and present some recent analytical and computational results in these areas. One of the results I'll discuss is a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet energies of the partition components. With intuition gained from an analogous continuous problem, we introduce a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations.