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Friday, November 30, 2018

**Abstract:** The notion of an infinitesimal quantity eluded rigorous treatment until the work of Abraham Robinson in 1960. Recent extensions and applications of his theory, called nonstandard analysis, have produced new results in many areas including operator theory, stochastic processes, mathematical economics and mathematical physics. 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 a 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. After an introduction to this powerful method, I will discuss applications to calculus, the imbedding of topological spaces into compact spaces, and measure and probability theory. This includes the work of Y. Sun who showed that the measure spaces introduced by the present speaker can be used to finally make sense of the notion of an infinite number of equally weighted, independent random variables in probability theory and economics.