Mathematics

On a class of oscillatory integral operators with fold singularities

Andreas Seeger (University of Wisconsin-Madison)

Groups, measure and the NIP, Section 6

Ayhan Gunaydin (UIUC Math)

Abstract: We continue the presentation of Groups, measure and the NIP where we left off on Friday, with Section 6, "Groups with NIP".

The Geometry of Hopf Maps

Richard L. Bishop (UIUC Department of Mathematics)

Abstract: The standard Hopf map from the 3-sphere to the 2-sphere has long been used to get topological information, specifically, homotopy groups of spheres. Less used is the rich geometry associated with the Hopf map -- in particular, the wealth of Clifford torii given by taking the inverse images of circles in the 2-sphere. The fact that they are of constant mean curvature is easily proved by using the relation of the group structure on the 3-sphere to the Hopf map. But there are some other 3-dimensional spaces with group stuctures, the simplest of which is Euclidean space, but more interestingly, the anti-de Sitter Lorentz space. Treating them analogously leads to some other Hopf maps, with consequent families of cylinders, which are constant-mean curvature homogeneous intrinsically- flat surfaces.

Random Geometrical Optics -- From Waves to Diffusion

Lenya Ryzhik   [email] (U Chicago Math)

Abstract: Kinetic equations are commonly used to describe multiple scattering of acoustic waves in a complex medium. However, there are very few rigorous results on the transition from the wave equation to the kinetic models. As an example where the diffusive limit may be obtained rigorously we consider wave propagation in the regime of random geometric acoustics. We show, starting from the wave equation that in the long time - large distance limit wave energy density behaves diffusively and identify the diffusion coefficient in terms of the statistics of the random medium. This is a joint work with T. Komorowski.

Nonstandard representations of Young measures, cont

David Ross (visiting UIUC from Hawaii 05-06)

Abstract: Young measures are measure-valued functions that arise as generalized limits of sequences of measurable functions with increasing oscillation. Nonstandard analysis allows for natural representations for both Young measures and the usual (narrow) topology on the space of Young measures.

Riemann Hypothesis for Varieties over Finite Fields

Shivi Bansal (UIUC Math)

Abstract: Deligne'e proof (1974) of the Weil conjectures is one of the greatest triumphs of modern algebraic geometry. These conjectures, inspired in part by the Riemann Hypothesis, relate the number-theoretic, algebraic and topological properties of a set of polynomial equations. I will explain what the conjectures say and the main tools invented or required for the proof. No prior knowledge of zeta functions or related topics will be assumed.

On hereditary combinatorial structures

Jozsef Balogh (UIUC Math)

Abstract: A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, words, permutations) that is closed under isomorphism and under taking induced substructures (like induced subgraphs) and contains arbitrarily large structures. Given a property P, we write P_n for the number of distinct (non-isomorphic) structures in P with n elements, and the sequence |P_n| is the speed of P. The speed of words was studied first by Morse and Hedlund in 1938. In the last few years, the ex-Stanley-Wilf conjecture (now the Klazar-Marcus-Tardos Theorem) was studied on the speed of permutations. In this talk I survey that area, pointing out generalizations toward ordered graphs, including some short proofs as well.

Mathematical Resonance Problems in Photonics

Michael I. Weinstein (Department of Applied Physics and Applied Mathematics, Columbia University)

Abstract: We describe the modeling and mathematical analysis of several problems arising in the study of photonic microstructures. This leads to interesting questions concerning dispersive Hamiltonian partial differential equations, having a structure analogous to particle/ field models in physics