| Date | Speaker | Title | Abstract |
| 01/25/2006 | Prof. Saugata Basu | Algorithms and Complexity in Real Algebraic Geometry. | I will give a brief survey of recent developments in the area of algorithmic real algebraic geometry, and describe the main open questions. No background in real algebraic geometry will be assumed. |
| 02/01/2006 | Prof. Ronghua Pan | Mathematical verification of Darcy law. | Darcy law is very important in characterization of the motion of waves in porous medium. It has been verified by experiments in physical point of view. I will explain how to verify Darcy law in mathematical point of view. Little PDE background is needed. |
| 02/08/2006 | Prof. Serge Guillas | Time series for the environment. | Time series analysis can be used to address environmental issues. Several case studies are discussed in this talk: ground-level pollution (ozone and sulfur dioxide), stratospheric ozone and climate change. First, we introduce the theory of time series, including Hilbert-valued ones. Second, we detail how statistical forecasting works. Third, we present open scientific problems, such as the detection of trends, that may be solved through a careful study of uncertainties: standard regression, Bootstrap methods, and some Bayesian approaches are succinctly described. |
| 02/15/2006 | Dr. Nathan Geer | Quantum group invariants of knots. | There are deep connections between quantum group theory and knot theory. At the heart of this connection is the Jones polynomial. In this talk I will give basic definitions relating to knot theory and discuss the Jones polynomial. If there is time I will give the construction of the Reshetikhin-Turaev quantum group invariant associated to the Lie algebra sl(2) and explain how this invariant relates to the Jones polynomial. |
| 02/22/2006 | Prof. Chris Heil | Frames, Time-Frequency Analysis, and Wavelets. | We will survey frame theory and some of its uses and application. A Frame is like a basis in that every element of a given space can be written as an (infinite) linear combination of the frame vectors. The frame vectors typically serve as simple "building blocks" from which complicated signals or functions can be built. However, unlike bases, frame decompositions are not unique in general. |
| 03/01/2006 | Prof. Evans Harrell | The extreme sport of eigenvalue hunting. | Energy levels in quantum-scale devices are the eigenvalues of some differential operators that depend on the shape of the device. As a rule the energy eigenvalues depend on the geometry in a complicated way, but if we seek the shape that maximizes or minimizes the energy, there is sometimes a simple answer. I'll describe a few cases with simple extrema, including some ``isoperimetric'' results, and how to get there with a blend of geometry, operator theory, and analysis. Some of this is joint work with Michael Loss and Pavel Exner. |
| 03/08/2006 | Prof. Thang Le | Local rules for quasi-crystals. | Quasi-periodic tilings of the Euclidean spaces can serve as models for real quasi-crystals. A local rule for a class of quasi-periodic tilings would correspond to local interactions that enforce the long-range order of the quasi-crystals. We will discuss local rules of the famous Penrose tilings, and the extistence of local rules for more general quasi-periodic tilings. |
| 03/15/2006 | Prof. Wilfrid Gangbo | Hamiltonian ODE's in the space of probability measures. | We consider a Hamiltonian H on P2(R2d), the set of probability measures with finite quadratic moments on the phase space R2d = Rd × Rd. This is a metric space when endowed with the Wasserstein distance W2. We prove existence of solution for the initial value problem dµt/dt+\nabla \cdot (\JJd vtµt)=0, where \JJd is the canonical symplectic matrix, µ0$ is prescribed, vt is a tangent vector to P2(R2d) at µt, and belongs to \partial H(µt), the subdifferential of H at µt. When H is \lambda--convex, proper and lower semicontinuous on P2(R2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system: H(µt) = H(µ0). Our study covers many systems occurring in fluid mechanics. (Joint work with L. Ambrosio) |
| 03/22/2006 | Spring Break | 03/29/2006 | Nathanael Berglund | Knitting as a means of Visualizing One-Sided Surfaces | What do mathematicians do for a little leisurely fun? Probably things like this! In this talk I consider knitting as a medium for visualizing one-sided (i.e. non-orientable) surfaces, particularly the Klein Bottle and Projective Plane. I discuss the various ways of representing these surfaces in R^3, along with their pros and cons from a visualization standpoint. Since we cannot hope to embed a non-orientable surface in R^3, I discuss why an immersion is the "next best" thing, and why knitting is especially well-suited to immersed surfaces! I will begin the talk by giving some of the basic topological theory of compact surfaces, and how thanks to a couple of theorems from differential topology (the Whitney embedding and immersion theorems) we may embed any smooth m-dimensional manifold in R^{2m} and immerse it in R^{2m-1}. I will then talk about the actual process of knitting such a surface, and the difficulties/challenges I encountered in my own attempt to produce a pattern for, and then actually knit, an immersion of P^2 known as "Boy's Surface". I will give some examples of how knitting and crocheting have been used by others to visualize one-sided and other surfaces, and as a way to illustrate the metric properties of certain surfaces. Finally, I will discuss directions for potential research into knitting of surfaces, such as automatic computer generation of stitch patterns for surfaces, which likely has connections with the problem of "ideal" texture mapping in computer graphics. This talk will be light on the mathematical side, being geared towards a general audience (and in particular anyone who likes to knit!). | 04/05/2006 | Prof. John McCuan | The variation of area, mean curvature, and open problems | I will describe the general formula for the first variation of area for surfaces and its relation to capillary surfaces and soap films. I will also describe several problems associated with capillary surfaces and soap films that are easy to state; some have been recently solved and some are still open. | 04/12/2006 | Prof. Vladimir Kolchinskii | Uniform Approximation in Laws of Large Numbers | Let $X, X_1, \dots, X_n$ be independent random points in a space $S$ with common distribution $P$ and let ${\cal F}$ be a class of real valued functions on $S.$ For $f\in {\cal F},$ denote by $Pf$ the expectation of $f(X)$ and denote by $P_nf$ the following average: $$ P_nf:=n^{-1}\sum_{i=1}^n f(X_i). $$ What is the size of the random variable $$ \xi_n({\cal F}):=\sup_{f\in {\cal F}}|P_nf-Pf|? $$ This question is very basic in many areas of Probability and Statistics. In the case when ${\cal F}$ consists of a single function $f,$ the answer is given by the classical Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm and related inequalities. In the case of infinite ${\cal F},$ the question is central in Probability in Banach Spaces. In the recent years, it has become clear that many mathematical problems in Theoretical Computer Science (namely, in Learning Theory) are also related to this question. We will discuss several situations in which the answer is known as well as some open problems. | 04/19/2006 | Yongfeng Li | Weak KAM theorem on Lagrangian dynamics | In the Hamiltonian systems, finding an invariant set by the Hamiltonian flow is the same as finding global solutions of the associated Hamilton-Jacobi equation. Such solution do not exist in general. KAM theory asserts that a Cantor set of global solutions exists when the Hamiltonian is a small C^k perturbation of a completely integrable Hamiltonian. While the weak KAM theorem of Albert Fathi asserts that such solutions always exist in a weak sense if the Hamiltonian or the associated Lagrangian is convex and superlinear. In this talk, a brief introduction to weak KAM theorem will be presented based on Fathi's notes about Weak KAM Theorem on Lagrangian Dynamics. | 04/26/2006 | Prof. Doron Lubinsky | Are Polynomials Lightweight? | We discuss a number of problems involving polynomials: how fast can we approximate by them, or by rational functions? When can they approximate "all" functions? What about approximating by Muntz polynomials, or on unbounded intervals? Finally, why is potential theory so useful in analyzing them? |