Departmental Colloquium

Numerical experiments on the ion acoustic plasma equations show the Korteweg deVries equation to be an extremely robust model: the qualitative features of the two soliton interaction are preserved even at large Mach numbers, where the KdV approximation is no longer quantitatively accurate. Collisions remain almost perfectly elastic, while the scattering shifts are larger than those in the KdV two soliton solution.

Zeta functions, and more generally L -functions, have long played an important role in number theory. While the classical L -functions attached to algebraic number fields continue remain to some extent quite mysterious, geometric methods have yielded much insight into L -functions attached to algebraic varieties defined over finite fields. In this talk we will discuss some of these geometric techniques and the consequent results concerning L -functions, and in particular explain a new result along these lines, due to the speaker and Mark Kisin, which applies to L -functions with p -adic coefficients in characteristic p.

In the mid-1980's, a small group of theoretical physicists studying string theory approached algebraic geometers with a curious proposition. Aside from the four classical space-time dimensions, our universe ought to have six dimensions curled up in a very small compact manifold V. Moreover, from considerations of (super)-symmetry, this manifold ought to have the following additional properties, placing it in the realm of algebraic geometry: V is a complex manifold with a Kähler form; V is simply connected; V admits a non-degenerate volume form. From these beginnings an extraordinary interaction has developed between string theorists and algebraic geometers, which I will attempt to paint in broad strokes.

In this talk we will discuss the Langlands conjecture for function fields (now a theorem of Lafforgue) and the geometric Langlands conjecture. Most of the lecture will consist of formulating and explaining these conjectures. We will also briefly discuss the proof of the geometric Langlands conjecture. This is joint work with Ed Frenkel and Dennis Gaitsgory. 4:30 PM in JFB 103

Free product in group theory, the non-abelian version of direct sum, has a geometric meaning in terms of fundamental groups. The question as to whether a given group can be written as a non-trivial free product has a few answers in specific cases; for instance, the fundamental group of a closed surface cannot be a non-trivial free product. By putting together results from the distant past, we can generalize this fact and ask more questions than give answers.

Every torus over a local field has a unique Neron model, defined over the ring of integers of that local field. Although this has been know for decades, only recently we begin to understand properties of this model and its consequences to number theory and representation theory. We will survey the known results, including the speaker's recent joint work with C.L. Chai about congruences of Neron models.

The definition of analytic-geometric category, introduced by van den Dries and Miller, extends the notion of subanytic sets and maps. Technically, it is a translation into geometric language of model-theoretic results of Willkie and Marker-McIntyre-van den Dries. After an introduction to these ideas, I shall argue that they provide a very useful tool.

In many situations in biology it is useful to consider a model that represents space as a grid of sites each of which can be in one of a finite number of states and changes at a rate that depends on the state of finitely many neighbors. Durrett and Levin proposed in 1994 that the behavior of these systems can be inferred by looking at the associated mean field ODE that is obtained by pretending that all sites are always independent. We will describe the answers that result from this approach for a number of systems of interest in biology and illustrate our results by a videotape of computer simulations.

Let X be a complex manifold. It is a fundamental fact that there is a correspondence between linear differential equations on X (i.e. vector bundles with flat connection) and complex representations of the fundamental group of X (i.e. local systems of complex vector spaces on X ). This correspondence is an equivalence of categories respecting direct sums, tensor products etc. One deficiency is that if f: X --> Y is a map of complex manifolds, the direct image of a vector bundle (resp. local system) need not be a vector bundle (resp. local system). To remedy this one generalises both types of objects: Vector bundles are replaced by D -modules, and local systems are replaced by constructible sheaves. Also, one now gets a correspondence not between objects, but between complexes - so a single D -module corresponds to a complex of constructible sheaves. This is called the Riemann-Hilbert correspondence, and it respects tensor product, and direct and inverse image functors. After reviewing the classical complex situation, I will report on joint work with M. Emerton concerning an analogue of this correspondence in characteristic p. These ideas have lead to a proof of an old conjecture of Katz on p -adic L -functions.

We have recently found that consideration of the differentiability properties of functions, analytic continuation, spectral decompositions, and various asymptotic expansions can be combined to yield highly accurate, fast algorithms for the solution of problems of scattering containing large two- or three-dimensional scatterers. Some of these numerical methods solve problems of scattering by penetrable bodies, others are concerned with rough surfaces, yet others deal with scattering problems by large, locally smooth conducting surfaces. In this talk I will discuss some of the main elements in these approaches, I will demonstrate their substantial accuracy with a variety of numerical results, and I will mention a number of applications - ranging from optical tomography in biological tissue to radar and remote sensing - for which such accuracy is highly desirable.

Work in collaboration with Oscar Bruno (CalTech). We study the macroscopic, mechanical behavior of composite materials consisting of many rigid, magnetically permeable, spherical inclusions embedded firmly in an isotropic, non-magnetic, continuous matrix. Such composites belong to a class of materials called magneto-rheological (MR) solids. The inclusions in MR solids are typically micron size iron, or iron based alloy (carbonyl-iron, iron-cobalt, etc.) particles. These particles are suspended in a non-magnetic elastomer like rubber, polymer gell, etc. Upon application of a magnetic field, the rheological properties of MR materials are rapidly and reversibly altered. The mechanism responsible for this bulk effect is the induced magnetic interaction between ferromagnetic particles in the composite. We consider the coupled elastic and magnetic problem in MR solids. We give a general framework for calculation of bulk properties of MR solids with randomly distributed ferromagnetic particles in volume fraction $\Phi$. For small volume fractions, we calculate the average stress-strain law in elastomer-ferromagnet composites, correct to order $\Phi^2$, by considering both elastic and magnetic interactions between pairs of particles. We show that, due to induced magnetic interactions, the bulk properties of the composite are altered significantly. Specifically, we show that the average strain depends on the elastic properties of the elastic matrix, the volume fraction $\Phi$ and the external magnetic field H _0. The average strain depends on the surface tractions, as well. However, this dependence is not a simple one, as in the problem of pure elasticity. We show that, for a free boundary (no surface tractions), magnetic interactions in the composite cause the material to self-deform. Depending on the softness of the matrix, the volume fraction and the strength of the external magnetic field, this deformation can be quite significant, as high as 10%. Furthermore, the deformation consists of an overall compression, although the average strain in the direction of the external magnetic field is different than the strain in directions orthogonal to H _0. We also show that, for nonzero surface tractions, the response of the composite depends strongly on the strength of the external magnetic field.