| Date | Speaker | Title | Abstract |
| 09/21/2005 | Doron Lubinsky | Orthogonalities, polynomial and other | Orthogonal polynomials appear in many contexts, and their asymptotics are widely used in mathematical physics. The study of these asymptotics is a hot topic now, with some groups using Riemann-Hilbert techniques, others using operator theory, and older groups sticking to Bernstein-Szego methods. We discuss some recent developments. There are other types of "polynomials" where orthogonality plays a role: Muntz polynomials, biorthogonal polynomials, ... . We discuss some recent activity in these areas too. |
| 09/28/2005 | Bill McClain | Two Dimensional Corners | We're finding a bound on the answer to the following question but stated in a finite field setting: How big must a subset of the square integer lattice {1,...,N} x {1,...,N} be in order to guarantee it has a corner? That is, a triple of points (x,y), (x+d,y) and (x,y+d) for some d > 0. |
| 10/05/2005 | William Green | Geometric Means of Positive Operators | We give a quick description of spectal theory for bounded normal operators. We then use this theory to show how to define the geometric and harmonic means of positive operators. When properly interpreted, many results for numbers (e.g., the arithmetic-geometric-harmonic mean inequality) remain true for positive operators. |
| 10/12/2005 | Federico Bonetto | Fourier's Law: a challenge for theorists | Since the creation of Statistical Mechanics more than a century ago, there have been many attempt to derive the law of heat conduction from the first principles of mechanics, without much success. We will try to review the model and methods involved in this research. |
| 10/19/2005 | Jeff Geronimo | Trigonometric polynomials on the bi-circle | The theory and applications of orthogonal polynomials in several variables is still quite undeveloped. I will describe a recent application related to factorization of positive trigonometric polynomials on the bi-circle. |
| 10/26/2005 | John Etnyre | What is Contact Geometry? | Contact geometry was born more than two centuries ago in the work of Huygens, Hamilton, Jacobi as a geometric language for optics. It was soon realized that it has applications in many other areas, including non-holonomic mechanics and thermodynamics. One encounters contact geometry in everyday life when parking a car, skating, using a refrigerator, or watching the beautiful play of light in a glass of water. Lie, Cartan, Darboux and many other great mathematicians devoted a lot of time to this subject. However, until very recently most mathematicians know little about contact geometry. In the last couple of decades contact geometry has taken a central place in low-dimensional topology and geometry. In this talk I will discuss some of the origins of contact geometry and hint at a few amazing recent developments. |
| 11/02/2005 | Torsten Inkmann | Introduction to Treewidth | The treewidth of a graph is a parameter that is important in Graph Theory and Theoretical Computer Science, but also has applications in other areas like Logic or Numerical Linear Algebra. The concepts related to treewidth are a key ingredient to the proof of one of the deepest results in Graph Theory, they are linked to the most important open problem in Theoretical Computer Science, and more recently, their usability in practice has been investigated. I will try to survey some of the main ideas related to treewidth and their impact in both theory and practice. The talk will be elementary; in particular all necessary concepts from Graph Theory and Computer Science will be defined. |
| 11/09/2005 | Leonid Bunimovich | Deterministic Walks in Random Environments | Deterministic walks in random environments (DWRE) occupy an intermediate position between purely random (generated by random trials) and purely deterministic (generated by deterministic dynamical systems) models. DWRE were (independently) introduced as phenomenological models in statistical physics, material science, communication theory, theory of artificial life, computer science, etc. where the corresponding moving objects were called particles, waves, signals, ants, read/write heads of Turing machine, etc. DWRE demonstrate many unusual (surprising) types of behavior from the point of a general intuition based on complete understanding of (some) purely probabilistic and purly deterministic systems. Recently some models of DWRE were explicitely solved which allowed to build some basic intuition on their behavior. The interest to DWRE (even more recently) increased when it was shown that some problems of the random matrix theory and of the quantum field theory can be reduced to DWRE. Analysis of DWRE requires a combination of the methods from Probability, Dynamical systems, Combinatorics and Topology. However, to attend (and even understand) this talk requires just to attend. |
| 11/16/2005 | Cinzia Elia | Using continuous and discrete SVD algorithms in dynamical systems | Stability spectra characterize the asymptotic behavior of non autonomous linear systems. The main techniques proposed to compute these quantities rely on the QR or on the SVD decomposition of the fundamental matrix solution of the system. In this talk, we first provide theoretical background on the Lyapunov and the Exponential Dichotomy spectrum on the half line. Then we describe continuous and discrete SVD techniques for the computation of these quantities. Our scope is twofold: understand under which assumption these computations are reliable and determine an efficient implementation for SVD techniques. |
| 11/30/2005 | Trevis Litherland | An Introduction to Longest Common and Longest Increasing Subsequence Problems | We will begin by introducing the classical longest common and longest increasing subsequence problems and will review some of the most important known results and open questions of the field. The main thrust of this talk will be to explore the natural ways in which typically probabilistic questions emerge as we study these problems. We will also indicate how these results are of significance to the bioinformatics community. |