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Seminars, Colloquia, and Conferences

2012 Norbert Wiener Lectures

Jordan S. Ellenberg

Professor of Mathematics, University of Wisconsin - Madison
Number theorist, Novelist, Blogger, Political Columnist


Public Lecture
There Is No Such Thing As Public Opinion: Polls, Hanging Chads, and Slime Molds
Tuesday March 27, 6-7pm, Cabot Auditorium, Cabot Intercultural Center
Reception: 7-8pm, Hall of Flags, Cabot Intercultural Center


Public opinion polls routinely show that large majorities of Americans support cutting spending and oppose raising taxes. But when lists of government programs are presented one by one, cuts in each program face majority opposition. What's going on here? A typical account is that Americans are irrational thinkers who want a free lunch, with low taxes and big government programs for all. The truth is more complicated. In fact, trying to put together the opinions of a heterogeneous population can lead to paradoxical results, even when the individuals involved are perfectly rational. The math that explains the puzzling polling on the budget -- first discovered by Condorcet in the midst of the French Revolution, and culminating in the Nobel-winning work of Kenneth Arrow -- also explains the vexingness of the Bush-Gore-Nader clash in Florida, and the apparently irrational decisions made by slime molds, primitive brainless creatures who biologists believe to be similar in certain respects to electorates.

Undergraduate Lecture
Polynomials as Numbers
Wednesday March 28, 4:30-5:30pm, Braker 1


It has been understood for a long time that there are deep analogies between polynomials and integers. By now, the study of that analogy has become a subject of its own. Ideas of number theory (like "prime numbers" and "zeta functions") carry over to polynomials, and ideas of polynomials (like "evaluating a polynomial \(P(x)\) at a value \(x\)" carry over to integers, and as a result our understanding of both areas is enriched and improved. I'll give an overview of this story, centering on counting questions: how many squarefree integers are there? How many squarefree polynomials are there? In what sense is the answer "the same"?

Arithmetic Counting Problems: The Topology of Numbers
Thursday March 29, 4:30-5:30pm, Bromfield-Pearson 101
Preceded by tea in the Mathematics Conference Room.


The study of class groups of number fields is one of the oldest parts of algebraic number theory, and is still almost entirely mysterious. The Cohen-Lenstra conjectures propose to understand the probability distribution obeyed by the class group of a "random" number field. These conjectures are not quite what you might expect: for instance, the probability that the class group has order not divisible by \(3\) is not supposed to be \(2/3\), as you might expect, but rather the infinite product \((1-\frac 13)(1-\frac 19)(1-\frac 1{27})\)..... I will explain recent work of myself, Akshay Venkatesh, and Craig Westerland which proves a version of the Cohen-Lenstra conjecture over the ring of polynomials over a finite field. In this case, it turns out that the Cohen-Lenstra conjecture in number theory is in fact reflecting a beautiful conjecture about topology -- in particular, the topology of certain moduli spaces called Hurwitz spaces. I'll explain our progress towards this conjecture and speak more generally about the relationship between topology and counting problems coming from arithmetic.

The Norbert Wiener Lectures were initially funded by an anonymous gift to the Department of Mathematics. All talks are free and open to the public.

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