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.

  • Sept 12
    Jerry Marsden, Control and Dynamical Systems, Caltech
    LAGRANGIAN REDUCTION AND ITS APPLICATIONS TO CONTROL
    Abstract: We shall focus on the mathematics of the reduction procedure for mechanicalsystems with symmetry from the point of view of Lagrangian mechanics. The general reduced Euler Lagrange equations will be discussed, along with the special cases due to Poincare and Routh. The general result relies on the use of connections and gauge theoretic view of mechanics. The generalization to systems with nonholonomic systems (such as rolling constraints) will be presented, and in particular, the momentum equation will be derived. The theory is applied to optimal control problems; the method of Lagrangian reduction provides a convenient alternative to the Pontryagin maximum principle for this class of systems. The special case of optimal control on Lie groups (important in problems like flight control) will be given special consideration.
  • Sept 26
    Sheldon Katz, Oklahoma State University
    THE SEARCH FOR A THEORY OF EVERYTHING: STRING THEORY AND MATHEMATICS
    Abstract: Throughout history, there have been numerous periods during which theoretical mathematics has interacted with other sciences to their mutual benefit. We are currently in the middle of such a period. The dialogue between string theorists and mathematicians began in earnest a little over ten years ago, and after a brief lull in the late 80's, the give-and-take has been rapidly accelerating. One of the great outstanding problems of theoretical physics is to unify the four basic forces of nature - the electromagnetic force, the weak nuclear force, the strong nuclear force, and the gravitational force. The first three are unified by quantum theory, and the last is explained by general relativity. There have been many efforts to unify these theories by developing a quantum theory of gravity. String theory is one approach; in fact it is arguably the only approach under study with a chance of being completely consistent. The benefits of the interaction between mathematics and string theory go both ways: 1. In building feasible models for string theory, physicists have needed to use theoretical geometric results developed by mathematicians over the last 150 years. Some of this geometric research is ongoing. 2. Physical models can be used to solve unsolved geometric problems. These answers to geometric problems stimulate new research in geometry, which in turn stimulate new physics, and so on. A survey of this interaction will be given.
  • Oct 17
    Ed Seidel, Relativity Group at the NCSA, UIUC
    SOLVING EINSTEIN'S EQUATIONS ON SUPERCOMPUTERS: GRAVITATIONAL WAVES, DISTORTED BLACK HOLES, AND BLACK HOLE COLLISIONS
    Abstract: Einstein's theory of gravity, general relativity, consists of 10 very complex, nonlinear partial differential equations for the gravitational field. The theory can be posed as a Riemannian 3-geometry evolving in time. Powerful new massively parallel computers have brought us to the brink of finding general numerical solutions to the Einstein equations for the first time since the theory was introduced in 1916. I will talk about recent efforts to develop axisymmetric and full 3D numerical codes to solve the Einstein equations for a number of interesting and important problems in pure gravitation and astrophysics, including gravitational waves, distorted black holes, and black hole collisions. Movies of some of these simulations, such as the collision of two black holes, will be shown.
  • October 31
    Klaus Schulten, Physics, UIUC
    TOPOLOGY REPRESENTING MAPS AND BRAIN FUNCTION
    Abstract: This lecture will address the question how the brain codes information to analyse a stream of sensory data in order to establish a suitable motor response. The main emphasis will be on visual control of grasping motions: How does a child learn to grasp simple cylindrical objects? How does the brain use a finite number of neurons and synaptic connections between them to code visual data and programs for hand motions? The development of a topologically optimized representation of information in the visual cortex of macaque monkeys will be reviewed and mathematically analyzed. A simple presentation of motor programs through affine maps will be introduced. The task of visuo-motor control will then be solved in the framework of an engineered camera-robot system employing a Delaunay tesselation for the space of visual data and a learning algorithm for locally affine maps.
  • Nov 7
    Phil Holmes, Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University
    KNOTS AND LINKS IN THREE DIMENSIONAL DYNAMICS
    Abstract: I will describe work due to Williams, Ghrist and myself on certain topological structures in three dimensional ordinary differential equations. A periodic orbit (or a collection of them) in a three dimensional phase space forms a knot (or link). Much is known topologically about knot and link structures, and the discovery of new polynomial invariants, such as those of Vaughn Jones (Fields Medal, 1990) has revitalised the area. However, the connections with differential equations and other areas of applied mathematics are relatively unexplored. I will show how knot and link invariants can be used to determine orbit genealogies and bifurcation structures in parameterised families of flows, and will describe a remarkable construction, due to Rob Ghrist, which reveals an ordinary differential equation containing ALL knots and links among its periodic solutions.
  • Nov 28
    Joseph Rosenblatt, Math, UIUC
    PHASE RETRIEVAL
    Abstract: All electro-magnetic radiation, from radio waves to X-rays, undergoes diffraction when passing through or around objects. If you want to reconstruct the object from the diffracted wave front, and can only measure the intensity of the diffraction, then you have to recover the phase of the diffraction in some manner. This phase retrieval problem is one of the central problems in X-ray crystallography, signal analysis, and other areas of optics where diffraction is studied. This talk will focus on the origins of the phase retrieval problem in physical chemistry, and the way the Fraunhofer limit arises from the Fresnel-Kirchoff diffraction formula. The problem of phase retrieval then becomes the problem of reconstructing a distribution from the modulus of its Fourier transform. Examples of non-uniqueness in phase retrieval, algebraic and analytical forms for a structure theorem for phase reconstruction, and some of the computational algorithms that arise inphase retrieval will be presented.