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for events the day of Thursday, December 5, 2013.

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Thursday, December 5, 2013

11:00 am in 241 Altgeld Hall,Thursday, December 5, 2013

Quadratic forms and the distribution of Fourier coefficients of half-integral weight modular forms

Maksym Radziwill (IAS Princeton)

Abstract: I will discuss recent joint work with Soundararajan concerning the distribution of Fourier coefficients of half-integral weight modular form. We obtain half of the conjecture concerning the log-normality of these coefficients. I will discuss applications to the number of representations of an integer by a quadratic form.

1:00 pm in Altgeld Hall 347,Thursday, December 5, 2013

Nielsen equivalence, stabilization and genericity

Ilya Kapovich (UIUC Math)

Abstract: We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction. This talk is based on a joint paper with Richard Weidmann; arXiv:1309.7458.

2:00 pm in 243 Altgeld Hall,Thursday, December 5, 2013

A direct proof of Gromov's Non-Squeezing Theorem

Alex Tumanov (UIUC Math)

Abstract: In his celebrated paper of 1985, Gromov developed a theory of J-complex (pseudoholomorphic) curves as a powerful tool in symplectic geometry. One of the headlines of that paper is the Non-Squeezing Theorem. It says that the unit ball B^(2n) in R^(2n), n>1, can be symplectically embedded in the "cylinder" rB^2 x R^(2n-2), only if the radius r of the base of the cylinder is at least 1. The original proof as well as more recent versions are quite involved. In this talk we present a simple direct proof that uses only the standard scheme for solving the Beltrami equation in one variable and the Schauder principle. This work is joint with Alexander Sukhov.

2:00 pm in 149 Henry Administration Building,Thursday, December 5, 2013

Apery numbers, hypergeometric functions and modular forms

Shaun Cooper (Massey University, New Zealand)

Abstract: The Aprey numbers were used by R. Apery in his proofs of the irrationality of $\zeta(2)$ and $\zeta(3)$, respectively. The generating functions for the sequences fang and fbng are analogues of 2F1 and 3F2 hypergeometric functions, and they can be uniformized by modular forms. These, and many other similar examples, will be surveyed.

3:00 pm in Altgeld Hall,Thursday, December 5, 2013

Ideals of Enveloping Algebras

Stefan Catoiu (DePaul University)

Abstract: The enveloping algebra U of a finite dimensional Lie algebra L is a non-commutative polynomial algebra whose indeterminates are the basis elements of L. The ideals of U can be presented by generators, and this gives rise to nice arithmetic and combinatorial properties. In particular, one can describe prime, maximal, primitive, or principal series ideals of U. I will show how this process works on small examples like sl2 and sl3.