When?Wednesdays 10-11, starting February 2006

Where?Alavi Commons Room

** April 7: ** Jeff Strom ** The Lusternik-Schnirelmann Theorem **

The Lusternik-Schnirelmann theorem relates a certain numerical invariant of manifolds to the number of critical points of a function f: M --> R. I'll prove this theorem and sketch an application to the existence of closed geodesics on smooth manifolds.

** March 31: volunteers sought! **

** March 24: ** Lixin Shen
** An Introduction to Wavelets, Part I **

The wavelet transform is a tool for carving up functions, operators, or data into components of different frequency, allowing one to study each component separately. The term wavelet was itself coined in 1982. In this talk, I will introduce the main idea of wavelets from a simple example, namely, Haar wavelets.

** March 17: ** Jim Zhu
** Variational methods and surjectivity, part 2 **

** March 10: ** Jim Zhu
** Variational methods and surjectivity **

Abstract.Variational methods are important tools in analyze surjectivity -- an important subject arise in many different settings including implicit function theorem, existence of solutions in (partial) differential equation theory and analysis of multifunctions. In the first part of the talk, I will use simple examples to illustrate ow to use variational methods to derive surjectivity results and introduce some basic concepts in convex analysis. Then, in the second part, I will present a recent concise proof of Rockafellar's celebrated result that in a rerflexive Banach space the sum of a maximum monotone multifunction and the duality map is surjective.

** March 3: ** No talk : Spring break!

** Feb 24: ** No talk at 11am.
Please note that Yuri Ledyaev presents colloquium at 4pm.

**Feb 17:** Lixin Shen
** Missing Data Recovery by Framelets **

Abstract:In this talk, I will present a missing data recovery algorithm based on framelets for three applications in image processing. These three applications are salt-pepper removal, random- valued impulse removal, and image inpainting.

**Feb 10:** Melinda Koelling
** Existence of Solutions to the Real Asymmetric
Toda Lattice **

Abstract:The Asymmetric Toda lattice is a solvable matrix differential equation. However, from the form of the solutions, it is not obvious for which initial conditions the solutions are nonsingular, i.e. exist for all time. We discuss a paper by Gekhtman and Shapiro that gives algebraic conditions on the initial condition that are equivalent to the existence of a nonsingular solution.

**Feb 3:** Yu.Ledyaev **Helly's Theorem on Manifolds **

Abstract:We discuss a generalization of the classical Helly's theorem on intersection of convex sets in euclidean finite-dimensional spaces for the case of manifolds of nonpositive curvature. This is a joint work with Jay Treiman and Jim Zhu in the framework of the NSF project "Nonsmooth analysis on smooth manifolds".