## Dr. Michael D. Weiss

7797 Heatherton Lane

Potomac, MD 20854-3264

Phone: 301-299-3144

E-mail: mdweiss@verizon.net

http://www.wolfram.com/services/training/weiss.html

Michael Weiss holds a Ph.D. in Mathematics from Brown University and an M.A. in economics from the University of Maryland. A former mathematics professor at Wayne State University, he served from 1976 to 1997 on the staff of the Economic Research Service of the U.S. Department of Agriculture, where his work emphasized the application of mathematics to economic theory and subjects related to agricultural economics and agriculture. He now serves as an independent consultant. His publications (both research and expository) include work on the economic theory of behavior under risk, nonlinear and chaotic dynamics in economics, the theory of fuzzy sets, topological and algebraic entropy, computer simulation of crop yield forecasting, and the use of symbolic mathematics software to simulate precision farming. As one of the mathematicians profiled in the MAA's "careers in mathematics" publications and as a past President of SIAM's Washington-Baltimore Section, he has a keen interest in communicating with students about nonacademic careers in mathematics as well as the many uses to which mathematics can be put.

Each of the following talks can be tailored to a wide variety of audiences.

**Mathematical Marvels: A Gentle Introduction to Modern Mathematics for Younger Students**

This talk is designed to bring the SIAM Visiting Lecturer Program to a new audience: students in grades 6-8+. Its aim is to instill an appreciation of modern mathematics through an early introduction to the beauty, fascination, and practical usefulness of this subject. The presentation uses a computer-projected display driven by Mathematica software, making it possible for students to see memorable mathematical images ranging from striking animations to dazzlingly colorful fractals to vivid examples of mathematically generated art. As an incidental benefit, students are given perhaps their first exposure to what modern mathematical software can do.

In a nontechnical and interactive manner, the talk presents thought-provoking glimpses of mathematical subjects that lie far beyond those normally discussed at this grade level. Among the topics considered are paradoxes of infinity, alternative forms of uncertainty (such as probability and fuzzy logic), computer simulation, fractals, chaos theory, and prime numbers. Throughout, I cite examples of how these and other mathematical ideas find application in such diverse fields as moviemaking, weather forecasting, and cryptography.

**Mathematics in Agricultural Economics: The Theory of Individual Behavior Under Risk**

How does a "rational" individual choose among risky alternatives? Expected utility theory, a creation of von Neumann and Morgenstern that has become the most widely used mathematical model of economic behavior under risk, represents risky situations as probability distributions and postulates that a rational individual chooses his or her "best" alternative by applying an individual preference ordering over a convex set of probability distributions. Since weather has a stochastic influence on crop production (and thus on farmers' incomes), portraying even such a seemingly "down-to-earth" decision as a farmer's choice of how much fertilizer to apply leads quickly to questions involving functional analysis over convex function spaces. In this talk, I will describe the mathematical foundations of individual choice under risk and will show how a rigorous approach to the subject can clarify such fundamental concepts as risk aversion.

**The New Dynamics: Nonlinearity, Chaos, and their Implications for Economics**

In recent years, research in both mathematics and the applied sciences has produced a revolution in the understanding of nonlinear dynamical systems. Used widely in economics and other disciplines to model change over time, these systems are now known to be vulnerable to a kind of "chaotic," unpredictable behavior. The slightest change in the initial conditions of a deterministic but nonlinear dynamic economic model can give rise to unpredictable changes in long-run behavior. Such a system can generate stochastic noise entirely endogenously, without the intervention of any external influence. In this talk, I will present the mathematical concepts underlying nonlinear and chaotic dynamics and will discuss their implications for economic modeling.

**Food, Farms, and Function Spaces: A Mathematical Approach to Problems of Agriculture**

In this talk, I present three examples of the application of mathematics, and especially probabilistic concepts, to agriculture. The first example shows how crop yield can be represented as a random surface (i.e., two-dimensional stochastic process, or random field). This characterization, which allows crop yield to be viewed as the Radon-Nikodym derivative of crop production, resolves the seeming paradox of portraying crop yield as both continuous and random over space.

The second example brings mathematics to bear on hamburgers! I describe binomial and Poisson probability models of firms' revenue risk in the face of a foodborne-disease outbreak that affects (for example) the independent outlets of a fast-food chain. Using a limit argument in which the probability of an outbreak in any one location approaches zero, I demonstrate that the proportion of its usual revenue that a firm can expect to lose is, in a certain approximate sense, directly proportional to the firm's size. Thus, larger firms are disproportionately at risk and, if rational, may try harder to avoid an outbreak.

The third example is an abbreviated version of my first talk, "Mathematics in Agricultural Economics: The Theory of Individual Behavior Under Risk."

**Using Symbolic Mathematics Software to Simulate Precision Farming**

Precision farming is a major new technology that allows farmers to create finely detailed maps, expressible mathematically as surfaces, that describe important characteristics of a farm field (such as fertilizer needs) by specific location. Computer-guided machinery keyed to such maps can assist a farmer in managing a farm field while responding to the field's spatially variable characteristics. Before the advent of precision technology, large farm fields had to be managed in a more spatially uniform manner.

A mathematical model of precision farming may draw upon such concepts as random surfaces (and their use in Monte Carlo simulations), operators mapping surfaces to surfaces, and functionals assigning numerical values to surfaces. This talk will describe the use of Mathematica symbolic mathematics software to implement these abstract ideas within a computer simulation model developed by the U.S. Department of Agriculture to examine some of the economic and environmental consequences of precision farming.