Mathematics in Science & Society
Fall 1999

Archive

Tuesday, September 14, 245 Altgeld Hall, 4:00 p.m.

Speaker: Walter K. Hayman, F.R.S., Imperial College of Science, Technology, and Medicine, London, England

Title: A Functional Equation Arising from Mortality Tables

Abstract: A law of mortality, proposed by Heligman and Pollard in 1980, and refining an empirical law observed by Gompertz in 1825, implies that beyond the age of 50, the odds of dying between age x and age x+1 increase exponentially with x. Two natural solutions of a functional equation arising from the Heligman-Pollard law are obtained. Among other results, it is shown that while the ratio of these solutions is an entire function with infinitely many zeros, on the real axis this ratio may be considered constant for actuarial purposes.

Tuesday, September 21, 314 Altgeld Hall, 3:00 p.m.
Coffee and cookies will be served at 2:30 p.m. in the Commons Room, 321 Altgeld Hall

Speaker: Professor Frank P. Kelly, Cambridge University

Title: Mathematical Modelling of the Internet

Abstract: Modern communication networks are able to respond to randomly fluctuating demands and failures by allowing buffers to fill, by rerouting traffic and by reallocating resources. They are able to do this so well that, in many respects, large-scale networks appear as coherent, almost intelligent organisms. The design and control of such networks present challenges of a mathematical, engineering and economic nature. This talk will describe how mathematical models are being used to address current issues concerning the stability and fairness of rate control algorithms for the Internet and for developing broadband networks.

Tuesday, October 12, 245 Altgeld Hall, 4:00 p.m.

Speaker: Anton Alekseev, Uppsala University, Sweden

Title: Noncommutative Geometry and D-branes

Abstract: The concepts of noncommutative geometry always played an important role in quantum physics: they provided a natural framework for noncommutativity of coordintaes and momenta, and for the Heisenberg's uncertainty principle. Recently, noncommutative geometry found new applications in the D-brane physics. In simple cases the D-branes can be viewed as self-adjoint boundary conditions for a certain second order differential operator A. Then, the noncommutativity results from the properties of the Green's function of A. More challenging examples are provided by D-branes on group manifolds. This talk is designed as an introduction into the subject. No previous knowledge of strings, D-branes, or noncommutative geometry is assumed.

Tuesday, October 26, 314 Altgeld Hall, 4:00 p.m.

Speaker: Christian Borgs, Microsoft

Title: Phase Transitions and Hardness: A Physics View of Algorithms

Abstract: Understanding the hardness of algorithms is a fundamental problem in mathematics and computer science. Since many algorithms used in practice are randomized algorithms, the study of such algorithms, in particular Monte Carlo algorithms, has become increasingly popular. In this talk I discuss how concepts from physics, in particular the theory of phase transitions, can help us to understand the hardness of certain algorithms, and how an insight into the phase structure of combinatorial models allows us to improve certain algorithms. No prior knowledge of algorithms or phase transitions is assumed in this talk.

Tuesday, November 2, 245 Altgeld Hall, 4:00 p.m.

Speaker: Avner Friedman, University of Minnesota

Title: Mathematics: the Unifying Factor in Science and Technology

Abstract: The talk will describe several recent examples where totally diverse problems in technology, industry and in the physical and life sciences have been successfully addressed by the very same mathematical ideas. One such example from industry involves detecting pollution in a diesel engine on one hand, and improving the quality of a photographic film on the other hand. Another example involves a model problem of tumor growth and a model of injection mold of viscous fluid.

Tuesday, November 30, 245 Altgeld Hall, 4:00 p.m.

Speaker: Francesco Fassò, University of Padova

Title: Stability of Elliptic Equilibria of Hamiltonian Systems: Some Recent Developments

Abstract: The talk illustrates some recent results on the stability properties of elliptic equilibria of Hamiltonian systems. The employed approach is based on the methods of Hamiltonian perturbation theory, specifically, on Nekhoroshev's theory, which is capable of giving stability results for finite, but extremely long, times. The interest of this kind of `practical stability' is in that it applies to equilibria which are Lyapunov unstable, but nevertheless, their instability manifests itself only on extremely long time scales: for practical purposes, they may be considered as stable.

From the technical point of view, the key notion is that of `directional quasi--convexity' of the fourth order Birkhoff normal form, which extends the `quasi--convexity' hypothesis of the standard Nekhoroshev's theory and appears to be a rather natural ans easily checkable hypothesis for the problem. Applications to two problems from Celestial Mechanics are described, namely, the stability of the equilateral Lagrangian of the restricted spatial three body problem and that of the Riemann ellipsoids.

Last modified November 4, 1999