Departmental Colloquium

The notion of a building is a simplicial complex introduced by Jacques Tits to study simple algebraic groups over arbitrary fields. We give a brief introduction to buildings and then describe some of their uses.

TBA

The inverse scattering problem arises in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. A general model is concerned with a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain. Given the incident field, the direct problem is to determine the scattered field for the known scatterer. The inverse medium scattering problem is to determine the scatterer from the boundary measurements of near field currents densities. Although this is a classical problem in mathematical physics, numerical solution of the inverse problem remains to be challenging since the problem is nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications. In this talk, our recent results in mathematical analysis and computational studies of the inverse boundary value problems for the Maxwell equations will be reported. A novel continuation approach based on the uncertainty principle will be presented. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse medium scattering problems. Convergence issues for the continuation algorithm will be examined. Our most recent progress on inverse source problems will also be discussed.

Harrison and Kreps showed in 1978 how the heterogeneity of investor beliefs can drive speculation, leading the price of an asset to exceed its intrinsic value. By focusing on an extremely simple market model -- a finite-state Markov chain -- the analysis of Harrison and Kreps achieved great clarity but limited realism. My talk discusses joint work with Xi Chen, which achieves similar clarity with greater realism by considering an asset whose dividend rate is a mean-reverting stochastic process. Our investors agree on the volatility, but have different beliefs about the mean reversion rate. We determine the minimum equilibrium price explicitly; in addition, we characterize it as the unique classical solution of a certain linear differential equation. Our example shows, in a simple and transparent manner, how heterogeneous beliefs about the mean reversion rate can lead to everlasting speculation and a permanent "price bubble".

While maybe as hard to parse, sadly it is not as catchy as Mark Twain's "Constantinopolitanischerdudelsackspfeifenmachersgesellschafft". In the 1930's I.M. Gelfand outlined a program of abstract harmonic analysis, which offered a paradigm for the use of symmetry to study a very wide class of mathematical problems. A key technical step is the following: Problem: For every locally compact group G, determine the set G^_u of irreducible unitary representations G. The group G is usually the symmetry group of a problem. In mathematical physics, differential geometry, or differential equations it is a real Lie group. In number theory, the group may be an algebraic group over a local fields. In combinatorics, it is often a finite group. In Gelfand's program, G is acting (as a symmetry group) on a measure space X, preserving the measure. In this setting there is a Hilbert space H = L^2(X) of (complex-valued) square-integrable functions on X. Then G acts linearly on H by [p(g)f](v):=f(g^{-1}v), and the fact that the action preserves the measure amounts to the fact that p(g) is unitary. The first step in Gelfand's program is to express questions about X (related to the symmetry group G) as questions about L^2(X) (and the linear operators p(g)). Knowledge of the unitary dual G^_u is a crucial ingredient. In this talk I will explain the nature of the answer of the unitary dual of a reductive real or p-adic group, and give some examples of its uses.