The theory of unitary representations originated in the past century as a natural evolution of classical Fourier analysis. In spite of the extremely significant contributions made by Langlands, Harish-Chandra and many other mathematicians, the problem of finding \emph{all} the unitary irreducible representations of a real reductive group remains a challenge: to this day, a complete answer is only known for a handful of groups. In this talk, I will describe some recent progress in the field.

Three different singularly perturbed eigenvalue problems in perforated domains, or in domains with perforated boundaries, with direct biological applications, are studied asymptotically. In the context of cellular signal transduction, a common scenario is that a diffusing surface-bound molecule must arrive at a localized signalling region, or trap, on the cell membrane before a signalling cascade can be initiated. In order to determine the time-scale for this process, asymptotic results are given for the mean first passage time (MFPT) of a diffusing particle confined to the surface of a sphere that has absorbing traps of small radii. In addition, asymptotic results are given for the related narrow escape problem of calculating the MFPT for a diffusing particle inside a sphere that has small traps on an otherwise reflecting boundary. The MFPT for this narrow escape problem is shown to be minimized for particular trap configurations that minimize a certain discrete variational problem (DVP). This DVP is closely related to the classic Fekete point problem of determining the minimum energy configuration for repelling Coulomb charges on the unit sphere. Finally, in the context of spatial ecology, a long-standing problem is to determine the persistence threshold for extinction of a species in a heterogeneous spatial landscape consisting of either favorable or unfavorable local habitats. For a 2-D spatial landscape consisting of such localized patches, the persistence threshold is calculated asymptotically and the effects of both habitat fragmentation and habitat location on the persistence threshold is examined. From a mathematical viewpoint, the persistence threshold represents the principal eigenvalue of an indefinite weight singularly perturbed eigenvalue problem, resulting from a linearization of the diffusive logistic model.

The analysis of these three PDE eigenvalue problems is based on the development of a common singular perturbation methodology to treat localized patches or traps in combination with some detailed analytical properties of the Neumann Green's function for the Laplacian. With this asymptotic framework, the problem of optimizing the principal eigenvalue for the each of these three problems is reduced to the simpler task of determining optimal configurations for certain discrete variational problems.

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Speaker: Michael Ward (UBC). Joint work with Dan Coombs (UBC), Alexei Chekhov (U. Sask), Alan Lindsay (UBC), Anthony Peirce (UBC), Samara Pillay (JP Morgan), Ronny Straube (Max Planck, Magdeburg).