Seminar Calendar
for events the day of Thursday, March 29, 2012.

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Thursday, March 29, 2012

Number Theory Seminar
11:00 am   in 243 Altgeld Hall,  Thursday, March 29, 2012
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Submitted by ford.
 Florian Luca (Univ. Morelia, Mexico)On the local behavior of the Ramanujan tau functionAbstract: Let $\tau(n)$ be the Ramanujan function given by $$\sum_{n\ge 1} \tau(n) q^n = q\prod_{i\ge 1} (1-q^i)^{24}.$$ Lehmer conjectured that  $\tau(n)\ne 0$ for all n, and this conjecture has been verified for all $n \le 22798241520242687999$: In my talk, I will present the main ideas of the following theorem obtained jointly with Yuri Bilu (Bordeaux): If $k$ is a positive integer such that $\tau(m) \ne 0$ for all $m \le k$, then for every permutation $\sigma$ of the first $k$ positive integers, there exist infinitely many positive integers $n$ such that $\tau(n + \sigma(1)) < \cdots < \tau(n + \sigma(k))$: In other words, the absolute value of the Ramanujan-function may behave in all possible ways (increasing, decreasing, etc.) on intervals of length $k$ provided of course that it does not vanish with a certain periodicity inside such intervals. The proof is quite elementary in essence (Chinese Remainder Theorem) although we appeal to lower bounds for linear forms in logarithms of algebraic numbers to justify certain estimates and to sieves and the truth of the Sato-Tate conjecture for $\tau(n)$ to avoid certain primes.

Group Theory Seminar
1:00 pm   in Altgeld Hall 347,  Thursday, March 29, 2012
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Submitted by kapovich.
 Matt Clay (Alleheny College)Relative twisting in Outer spaceAbstract: The Culler-Vogtmann Outer space is the space of marked metric graphs of a fixed rank. It plays a similar role in the theory of the group of outer automorphisms of a free group as the Teichmueller space of a surface plays for the mapping class group of the surface. I will discuss a tool for providing a lower bound on the distance between points in the Outer space with the Lipschitz metric that is akin to annular projection in surfaces. This is joint work with Alexandra Pettet.

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, March 29, 2012
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Submitted by aimo.
 Dean Baskin (Northwestern)Asymptotics of radiation fields in asymptotically Minkowski spaceAbstract: Radiation fields are (appropriately rescaled) limits of solutions of wave equations along light rays. In this talk I will describe a class of (non-static) asymptotically Minkowski space times for which the radiation field is defined and indicate how methods of Vasy can be used to express the asymptotics in terms of the resonances of a related Riemannian problem on an asymptotically hyperbolic manifold. In particular, even on Minkowski space, these methods give a new understanding of the Klainerman-Sobolev estimates. This is joint work with Andras Vasy and Jared Wunsch.

2:00 pm   in 241 Altgeld Hall,  Thursday, March 29, 2012
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Submitted by lukyane2.
 Santiago Camacho (UIUC Math)Some first order definable properties of Tropical GeometryAbstract: There are many different ways one can approach tropical geometry, approximation by amoebas generated by algebraic varieties in $\mathbb{C}^n$, the valuation map over algebraic varieties of general algebraically closed valued fields, or just simply the geometry of the min-plus tropical semiring of $\mathbb{R}$. On this talk we focus on these last two and give a sketch of a proof of the fundamental theorem of tropical geometry linking them together, using elementary model theoretic tools. No previous knowledge on logic will be a assumed.

Mathematics Colloquium
4:00 pm   in 245 Altgeld Hall,  Thursday, March 29, 2012
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Submitted by kapovich.
 James Haglund (University of Pennsylvania)Macdonald Polynomials and the Hilbert Series of the Quotient Ring of Diagonal CoinvariantsAbstract: Macdonald polynomials are symmetric functions in a set of variables X which also depend on two parameters q,t. In this talk we describe how a formula of Haiman for the Hilbert series of the quotient ring of diagonal coinvariants in terms of Macdonald polynomials implies a much simpler expression for the Hilbert series involving matrices satisfying certain constraints.