Schedule of talks for Spring, 1995

The talks are presented on Tuesdays at 4 PM in room 241, Altgeld Hall. (Note room change again.) Refreshments are served at 3:15 in room 321.

  • Feb 14
    Nicholas Yannelis, Department of Economics
    Equilibrium Theory with Incomplete Information
    Abstract: We examine the contracts made by traders who are differentially (privately) informed. In particular, we address the following questions. How do coalitions of traders share their private information? How can one measure the information superiority of a trader? Do traders have an incentive to report their true private information? Can one find an equilibrium in a market with privately informed traders?
  • Feb 21
    Thomas Huang, Department of Electrical and Computer Engineering, Beckman Institute
    Polynomial Methods in Computer Vision
  • Feb 28
    David K. Campbell, Department of Physics
    Sawtooth Circle Maps: Solvable Models for Mode-locking from Clocks to Cardiac Arrhythmias
    Abstract: To develop a deeper analytic understanding of the phenomenon of "mode locking" or "resonance" for nonlinear oscillators, we formulate and study two distinct parametrized families of piecewise linear approximations to the classic "sine circle" map. For these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. We examine analytically the phase-locked regions of arbitrary rational winding number, p/q, and show that these regions -- the analogues of the familiar "Arnold tongues" of the sine circle map -- are more appropriately termed "Arnold sausages" in the sawtooth maps, since they are characterized by having widths that shrink to zero at certain non-zero values of strength of the nonlinearity BELOW the "critical line." We also describe (and provide references to our proofs of) a number of theorems determining (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation intervals; and (3) the structure and bifurcations of the attractors (in one of the families). We close with some conjectures on the nature of the transition between piecewise linear and smooth circle maps.
  • Mar 7
    Dimitri Mihalas, Department of Astronomy
    Adaptive Grid Methods in Radiation Hydrodynamics
  • Mar 21
    Andrea Bertozzi, University of Chicago, Department of Mathematics
    The mathematics of moving contact lines
    Abstract: The spreading of a liquid on a solid surface occurs in many physical problems ranging from the formation and protection of microchips to the build-up of ice on an airplane wing. A useful mathematical setting for the study of spreading liquids is the ``lubrication approximation'' which, in the surface tension driven regime, yields a fourth order degenerate diffusion equation for the film thickness. There is a singularity associated with the `no slip' boundary condition for a moving contact line, the triple juction of the air/liquid/solid interfaces. Hence, modelling the dynamically elvolving contact line raises interesting questions involving the physics of different length scales. The physical singularity corresponds to a critical exponent in the fourth order PDE. Withing the context of the ``lubrication approximation'' I discuss two models for removing the singularity at the contact line for the case of `complete wetting' (1) using a slip condition on the liquid/solid interface and (2) removing the singularity with a `porous media' cutoff of long range Van der Waals interactions.'' Both mathematical theory and numerical simulations will be discussed.
  • Apr 11
    Hassan Aref, Department of Theoretical and Applied Mechanics
    Dynamics of discrete vortices
    Abstract: The lecture will primarily discuss the intriguing dynamical system, known as the point vortex equations, that captures essential elements of the interaction of strong vortices in two dimensions. I will indicate the physical assumptions that lead to this model system, and point out some of the mathematical problems to which it gives rise. The presentation will highlight the many curious twists that have occurred in the historical development of this subject. I will also allow myself some philosophical comments on the general topic of "applied mathematics" stimulated by the technical material under consideration.
  • Apr 18
    Nicholas Pippenger, University of British Columbia, Department of Computer Science
    Self-Routing Networks
    Abstract: During most of the twentieth century, interconnection networks such as those found in telephone exchanges have been controlled by centralized mechanisms, most recently by general-purpose digital computers. Once upon a time, however, such networks were controlled by auxiliary components distributed among the switching components. While the centralized approach offers greater flexibility, the distributed approach has the advantage in expandability and tolerance of overloads and failures. This prompts the study of self-routing networks for solving various communication tasks. Perhaps the earliest examples of self-routing networks are the comparator-based sorting networks that have been studied since the 1950's. In these networks the routing information passes unidirectionally through the network in the same direction as the signals. More recent work has produced examples of networks in which the control information passes both forwards and backwards, so that the network as a whole operates as a distributed computer in solving its own routing problems. Surprisingly, the pursuit of this study has led to the discovery of routing algorithms that are more efficient in some respects than the best serial algorithms that were previously known.
  • Apr 25
    Michael Stone, Department of Physics
    Kac-Moody Lie Algebras and the Quantum Hall Effect
    Abstract: I will describe the quantum Hall effect, and explain how the physics of the effect is woven from a number of mathematical strands. Although I hope to tease out and exhibit a number of these strands, my principal aim will be to show how the electrons in a quantum Hall device provide a concrete representation for the simplest affine (Kac-Moody) Lie algebras. No previous knowledge of these algebras will be required of the audience.
  • May 2
    Bruce Hajek, Department of Electrical and Computer Engineering
    Dynamic Load Balancing and Equilibria
    Abstract: Load balancing algorithms and the resulting equilibria are considered. There are overlapping sets of locations, and each set of locations has an associated load which is constrained to be allocated among the locations in the set. The total load at a location is thus the sum of the contributions due to the sets that contain it. Dynamic load balancing is a process whereby a load, arriving at a time-varying and possibly random rate, is allocated with the objective of promoting balance. The performance of a simple dynamic load balancing algorithm is examined for finite networks using fluid, diffusion and large deviation analysis. The work is motivated by the problem of dynamic allocation of resources in both wireline and wireless communication networks. Equilibrium is said to hold if the load corresponding to any one set cannot be reassigned to improve the balance of total loads. In this talk, the set of possible equilibria, or balanced load vectors, is examined for infinite networks. The concept of load percolation is introduced and is shown to be associated with infinite sets of locations with identical load. This talk is based in part on joint work with Murat Alanyali.