Bill Sears Colloquia - Spring 2001

All talks are Wednesdays 12:30-1 p.m. in 657 Rhodes Hall unless noted otherwise. Pizza is available at 12:10 p.m. if ordered before 10 a.m. on the day of the seminar. (To order pizza, send e-mail to dap7@cornell.edu.)

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• February 28 - Steve Vavasis, Computer Science, "Accurate Solution of Polynomial Equations by Eigenvalue Methods" (joint work with G. Jonnson)
  Abstract: Solving polynomial equations is one of the oldest problems in numerical analysis. About 100 years ago, Macaulay showed that solution of a system of polynomial equations can be reduced to computation of eigenvectors of a matrix. In this talk I'll describe Macaulay's algorithm, explain what goes wrong with it when implemented in computer arithmetic and how to fix it.
• April 4 - Ron Elber,, Computer Science, "Long Time Dynamics of Biomolecules"
  Abstract: Proteins are the basic machines of life. Once all the proteins are sequenced and their structures are determined, there remains the question of how these molecules work. A popular approach to simulate the function of proteins is the Molecular Dynamics (MD) technique. In MD we solve numerically initial value problems (Newton's equation) in small time steps. The equations are stiff which limits the solution to relatively short times. We are limited today to about 10E-9s of simulation time, while the time scales of many important biological phenomena are significantly longer. For example, protein folding takes milliseconds (10E-3s) and protein activation can take microseconds (10E-6s).
  I will discuss an alternative computational method that we developed, which is based on an optimization of a functional and solving boundary value problems. I will argue that with the functional optimization a much larger time step can be used, leading to automated filtering of high frequency motions. The result is an approximate (but numerically stable) trajectory. I will also argue in favor of integrating the equations of motion as a function of path length instead of time. The algorithm is easy to parallel and an implementation on the Theory Center Velocity clusters will be described.
• April 9 - Stephen Wicker, Electrical and Computer Engineering, "Game Theory and the Design of Self-Configuring, Adaptive Wireless Networks"
  Abstract: To realize a truly scalable wireless network, one must place the bulk of the decision-making burden on individual terminals or local clusters of terminals. Basic control and management functions must be distributed and local. In this talk I show how game theory can be used to develop local decision rules that lead to desirable, emergent global network performance. I begin with a very brief introduction to game theory, with a focus on the development of utility functions and Nash equilibria. I then apply these notions to two standard problems in wireless networks: power control and random access. I show that the tools of game theory lead to strategies in which optimal behavior emerges ``naturally'' from the selfish interests of the agents and the rules of the games.
  Much of the original work described in this talk was conducted by my PhD. student Allen B. MacKenzie.
• April 18 - Evan Cooch, Natural Resources, "Periodic Projection Matrices (or - why I should have taken that class in linear algebra)"
  Abstract: As the Greek philosopher Heracleitus noted (one or two years ago), 'All is flux, nothing is stationary'. The fact that natural systems are often (perhaps invariably) characterized by 'variation' is both intriguing (intellectually) and a considerable challenge (analytically). When considering models of population dynamics, we consider variation among the individuals in the population in particular vital rates (reproduction, survival), in the environment the individuals reside in, and in the interaction of the two.
  For many populations, where breeding events occur in discrete time, models are based on projection matrices, which are often constructed (and used) under assumptions (particularly) of time-invariance in the vital rates. While the analysis of such deterministic models is familiar (at least to most biologists), temporal variability raises significant new problems: what is the growth rate? What are the ergodic properties? These questions are extremely relevant to very practical concerns. A brief introduction to one approach to 'dealing with variation' - periodic matrices - will be discussed: some simple examples, and some questions for future work.
• April 25 - Thomas Coleman, Computer Science, "Computing a Smooth Volatility Function*"
  Abstract: Stock options are often priced, and hedged, using what is known as a 1-factor continuous diffusion model. This model requires, as input, a measure of the stock volatility (or variance). Volatility plays a crucial role in the accurate pricing (and hedging) of stock options. We discuss a recent method for computing a smooth local volatility function, or surface, in a 1-factor continuous diffusion. Given European market data, a tractable inverse problem is formulated to yield an OEimplied¹ volatility function. Our method combies the use of numerical optimization methodology and spline approximations. Finance background will not be required to follow (and appreciate) this talk ­ the underlying finance modeling ideas will be introduced as required.
  * Based on work with Cornell colleagues Yuying Li , Arun Verma, and Yohan Kim.