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for events the day of Friday, November 30, 2018.

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Friday, November 30, 2018

3:00 pm in Altgeld Hall 145,Friday, November 30, 2018

Infinitesimals in Analysis, Topology, and Probability

Peter Loeb (Illinois Math)

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.

4:00 pm in 241 Altgeld Hall,Friday, November 30, 2018

Counting curves in the plane

Nachiketa Adhikari (UIUC)

Abstract: There is a unique line through two points in the plane. There is a unique conic through five points (in general position). There are twelve cubics through 8 such points. So: is there a general formula for the number $N_d$ of degree d curves passing through 3d-1 points in the plane? In the 1990s, an astonishing relationship between invariants obtained in string theory and certain spaces of curves was discovered. Using these, Kontsevich obtained a recurrence formula for $N_d$. I will sketch a few of the (mathematical) ideas. This talk should be accessible to beginning graduate students.

4:00 pm in 345 Altgeld Hall,Friday, November 30, 2018

Solution of Christensen's problem on universally measurable homomorphisms

Christian Rosendal (UIC Math)

Abstract: Answering a longstanding problem originating in J.P.R. Christensenís seminal work on Haar null sets, we show that a universally measurable homomorphism between Polish groups is continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo ZF+DC, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on Cantor space has finite chromatic number.