Seminar Calendar
for events the day of Tuesday, September 27, 2011.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, September 27, 2011

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by jathreya.
 Jayadev Athreya (UIUC)Cusp Excursions for HorocyclesAbstract: In joint work with Yitwah Cheung, we obtain results describing the sequence of cusp excursions for horocycles on the modular surface. In the course of this work, we resolve a question of Boca-Zaharescu that arose in their study of Farey fractions. This talk will also be of interest to number theorists, and is in some sense a follow up to my number theory seminar on September 1.

Topology Seminar
11:00 am   in 241 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by franklan.
 Greg Chadwick (Indiana University)Structured Orientations of Thom SpectraAbstract: The Thom spectra originating in the study of cobordism are fundamental to stable homotopy theory. An understanding of such structured ring spectra is relevant to many key problems. For the complex cobordism spectrum MU, work of Quillen describes ring maps out of MU in terms of complex orientations. When the target is a more structured ring spectrum, it is natural to ask which maps preserve the multiplicative structure. In several interesting cases, such structured complex orientations live in the unit spectrum cohomology of a cover of the classifying space BU. For E2 or E4 maps this cohomology may be computed and demonstrates that every self ring map of MU is E2. This then shows the Brown-Peterson spectrum BP admits an E2 ring structure.

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by vzh.
 Maxim Arnold (UIUC Math)On products of skew rotationsAbstract: In recent problems of population genetics it appear some processes having the form of two interacting Hamiltonian systems. I shall discuss first trivial step in the understanding of arising dynamics and try to explain how our result can be generalized to the case of stochastic switching.

Number Theory
1:00 pm   in 241 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by berndt.
 Khang Tran (UIUC)Connections Between Discriminants and the Root Distribution of Polynomials with Rational Generating FunctionsAbstract: Let $H_m(z)$ be a sequence of polynomials and $D(z; t)$ be the denominator of its generating function $\sum_{m=0}^{\infty}H_m(z)t^m$. We show that in some cases, the roots of $H_m(z)$ are dense as $m\to\infty$ on some fixed arcs whose equations are explicitly given and whose endpoints are the roots of $Disc_tD(z; t).$ The proofs involve the $q$-analogue of discriminant, a concept introduced by Mourad Ismail.

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by sabok.
 Marcin Sabok (UIUC)Canonical Ramsey theory on Polish spaces (continuation)Abstract: Title: Canonical Ramsey theory on Polish spaces Abstract: A typical problem in Ramsey theory on a Polish space $X$ looks as follows: given an analytic subset $E\subseteq [X]^2$ and a $\sigma$-ideal $I$ on $X$, find a Borel set $B\notin I$ such that $[B]^2\subseteq E$ or $[B]^2\cap E=\emptyset$. If $E$ is an equivalence relation, this means that on the set $B$ the relation $E$ is one of the two trivial ones: the identity or everything. An easier problem: given an analytic equivalence relation $E$ on $X$, find a Borel set $B\notin I$ such that $E\restriction B$ is strictly simpler in the Borel-reducibility order than $E$ (e.g. smooth when $E$ is not smooth itself). There are many classical result of this kind which deal with particular cases of $\sigma$-ideals or analytic relations. A more uniform approach can be realized in the language of definable proper forcing. Given a Polish space $X$ and a $\sigma$-ideal $I$ on it, we study the quotient forcing $P_I$ of all Borel sets $B$ in $X$ which do not belong to $I$. The forcing properties of $P_I$ often depend on the analytical and descriptive properties of the $\sigma$-ideal $I$, but the common feature of most of the interesting cases is that the forcing $P_I$ is proper. It turns out that definable proper forcing can be successfully used in the study of Borel equivalence relations and Ramsey theory on Polish spaces. In this talk, I will discuss recent result in this area. Since we have the two factors that come into the play: the $\sigma$-ideal $I$ and the analytic relation $E$, the appropriate Ramsey theorem will be true only under specific assumptions on $I$ and $E$. This is joint work with Jindrich Zapletal and Vladimir Kanovei.

Probability Seminar
2:00 pm   in 347 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by kkirkpat.
 Kay Kirkpatrick   [email] (UIUC Math)Bose-Einstein condensation and a phase transition for the nonlinear Schrodinger equationAbstract: Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss work with Sourav Chatterjee about a phase transition for invariant measures of the discrete focusing NLS. Using techniques from probability theory, we show that the thermodynamics of the NLS are exactly solvable in dimensions three and higher. There are a number of consequences of this result, including a prediction for experimentalists to look for a new spatially localized phase of BEC.

Geometry Seminar
2:00 pm   in 243 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by clein.
 Yury Ionin (Central Michigan University Math)Splitting complete graphs into trianglesAbstract: Let the vertices of a complete graph $K_n$ be points in a Euclidean space, let no three of them be collinear, and let the segments connecting the vertices be regarded as the edges of the graph. Is it possible to partition the set of all edges into triangles? The answer to this question was obtained ca. 1850: it is possible if and only if $n \equiv 1$ or $3$ $(mod 6)$. In this talk we will consider a similar question with triangles replaced by triangular triples: three edges of $K_n$ form a triangular triple if their lengths satisfy the (strict) triangle inequality. We will show that if $n \equiv 0$ or $1$ $(mod 3)$, then the set of all edges can be partitioned into triangular triples; if $n \equiv 2$ $(mod 3)$ and one edge of $K_n$ is deleted, then the set of the remaining edges can be partitioned into triangular triples. This is joint work with Gregory Galperin, Eastern Illinois University.

Graph Theory and Combinatorics
3:00 pm   in 241 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by west.
 Matthew Yancey (UIUC Math)Large rainbow matchings in large edge-colored graphsAbstract: A rainbow matching in an edge-colored graph is a matching in which the edges have distinct colors. The color degree of a vertex $v$ is the number of distinct colors on the edges incident to $v$. Let $\hat\delta(G)$ be the minimum color degree among the vertices of $G$. We present several anti-Ramsey results about $rm(G)$, the size of the largest rainbow matching in $G$. We prove the conjecture by Wang and Li that $rm(G)\ge\hat\delta(G)/2$ when $\hat\delta(G) \geq 4$. We further show that $rm(G)\ge\hat\delta(G)$ when $|V(G)| > 18 \hat\delta(G)^3$. Both results are sharp. (This is joint work with Prof. Kostochka)

Mathematics Colloquium: Coble Memorial Lectures
4:00 pm   in 314 Altgeld Hall,  Tuesday, September 27, 2011
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Submitted by seminar.
 Ngo Bao Chau (University of Chicago)Geometry of the Hitchin fibrationAbstract: Nigel Hitchin observed that the cotangent bundle of the moduli space of principal bundles on a Riemann surface has a structure of an algebraic completely integrable system. A deep relation between the geometry of the Hitchen fibration and the geometry of the trace formula has been discovered. This has been helpful in the proof of the fundamental lemma in the Langlands program. The Coble Memorial Lectures will be held September 27-29, 2011. Before each lecture, coffee and cookies will be served in the Common Room (321 Altgeld Hall) at 3:30 p.m. A reception will be held on Tuesday, September 27, from 5-6 p.m. in 314 Altgeld Hall.