Peter Loeb (Department of Mathematics, University of Illinois)
Infinitesimals in Analysis and Probability Theory
Abstract: The notion of an infinitesimal quantity has been used in mathematics for over 2200 years. It eluded rigorous treatment until the work using model theory of Abraham Robinson in 1960 established a rigorous foundation for the use of infinitesimals in mathematics. Recent extensions and applications of his theory, now called nonstandard analysis, have produced new results in many areas including operator theory, stochastic processes, mathematical economics and mathematical physics. In all of these areas, infinitely small and infinitely large quantities can play an essential role in the creative process. At the level of calculus, the integral can now be correctly defined as the nearest ordinary number to an infinitely large sum of infinitesimal quantities. In Probability theory, Brownian motion can now be rigorously parameterized by a random walk with infinitesimal increments. In economics, an ideal economy can be formed from an infinite number of agents each having an infinitesimal influence on the economy. Spaces formed with nonstandard analysis give the simplest probability spaces for a continuum of independent random variables or traders in an economy. The talk gives an introduction to this fruitful area of mathematics. |
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