Seminar Calendar
for events the day of Tuesday, October 21, 2008.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 21, 2008

Topology seminar
11:00 am   in 241 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by mando.
 David Spivak (University of Oregon)Derived smooth manifoldsAbstract: Given a smooth manifold $M$ and two submanifolds $A,B\subset M$, the intersection $A\cap B$ need not be a smooth manifold. By Thom's transversality theorem, one can deform $A$ to be transverse to $B$ and take the intersection: the result, written $A\pitchfork B$, will be a smooth manifold. Moreover, if $A$ and $B$ are compact, then there is a cup product formula in cobordism, integral cohomology, etc. of the form $$[A]\smile [B]=[A\pitchfork B],$$ where $[-]$ denotes the cohomology fundamental class. The problem is that $A\pitchfork B$ is not unique, and there is no functorial way to choose transverse intersections for pairs of submanifolds. The goal of the theory of {\em derived manifolds} is to correct this defect. The category of derived manifolds contains the category of manifolds as a full subcategory, is closed under taking intersections of manifolds, and yet has enough structure that every compact derived manifold has a fundamental class. Even if $A,B\subset M$ are not transverse (in which case their intersection can be arbitrarily singular), their intersection $A\times_MB$ will be a derived manifold with $[A\times_MB]=[A\pitchfork B]$, and thus satisfy the above cup product formula. To construct the category of derived manifolds, one imitates the constructions of schemes, but in a smooth and homotopical way. I will begin the talk by explaining this construction. Then I will give some examples and discuss some features of the category of derived manifolds. I will end by sketching the Thom-Pontrjagin argument which implies that compact derived manifolds have fundamental classes.

CR Geometry Seminar
12:00 pm   in 345 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by jlebl.
 Alexander Tumanov (Department of Mathematics, University of Illinois)Analytic continuation from a family of linesAbstract: Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables. The proof is based on CR functions.

Harmonic Analysis and Mathematical Physics
1:00 pm   in 347 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by laugesen.
 Mark Ashbaugh   [email] (University of Missouri at Columbia)Universal Inequalities for Eigenvalues of Differential OperatorsAbstract: Beginning with Payne, Polya, and Weinberger in the mid-50's, many universal inequalities for the eigenvalues of various classical differential operators have been derived. Several of these take the form of inequalities for eigenvalue ratios. This talk will present some of the history, giving a flavor of the techniques, and will go on to present some of the more recent results.

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by ssolecki.
 Lou van den Dries (Department of Mathematics, University of Illinois)O-minimal triangulation respecting a standard part mapAbstract: Let $R$ be an o-minimal field and $V$ a proper convex subring of $R$ with ordered residue field $k$ and standard part map (residue map) $st: V \to k$. Let $k_ind$ be the expansion of $k$ by the relations $st(X)\subseteq k^n$ where $X\subseteq R^n$ is definable in $R$. When is $k_ind$ o-minimal? In her thesis Jana Marikova answers this question positively if $(R,V)$ satisfies a certain first order scheme $\Sigma$. More precisely: if $(R,V)\models \Sigma$, then the definable relations of $k_ind$ are exactly the boolean combinations of the basic relations $st(X)$. In subsequent joint work with Marikova we proved the converse of the above statement. The main tool is a triangulation result respecting the standard part map that seems to be new even in the semialgebraic case.

Probabilty Seminar/Stochastic and Non-linear Analysis seminar
2:00 pm   in 345 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by rsong.
 Prof. Joseph Conlon (University of Michigan)The Becker-Doering (B-D) and Lifschitz-Slyozov-Wagner (LSW) EquationsAbstract: The B-D equations describe a mean field approximation for a many body system in relaxation to equilibrium. The two B-D equations determine the time evolution of the density c(L,t) of particles with mass L, L=1,2,... One of the equations is a discretized linear diffusion equation for c(L,t), and the other is a non-local constraint equivalent to mass conservation. Existence and uniqueness for the B-D system was established in the 1980's by Ball, Carr and Penrose. Research in the past decade has concentrated on understanding the large time behavior of solutions to the B-D system. This behavior is characterized by the phenomenon of "coarsening", whereby excess density is concentrated in large particles with mass increasing at a definite rate. An important conjecture in the field is that the coarsening rate can be obtained from a particular self- similar solution of the simpler LSW system. In this talk we shall discuss the B-D and LSW equations, and some recent progress by the speaker and others towards the resolution of this conjecture.

Geometric Potpourri Seminar
2:00 pm   in 243 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by j-wetzel.
 Profs. Gregory Galperin and Andrew Mertz (Department of Mathematics and Computer Science, Eastern Illinois University)The complexity of finite sequences and geometry of finite vector spacesAbstract: Leibnitz considered "monads," the simplest mathematical object that can be imagined: a map of a finite set M to itself, together with its graph. It turns out that each connected component of the graph has a unique cycle to which different kinds of trees are attached. If M is the set of vertices of the d-dimensional unit cube (all coordinates of which are 0's and 1's), the map can be defined, as was done by Newton, to be the differences of the binary coordinates modulo 2. In our talk we will show many interesting (and sometimes unexpected) properties of the graph in this case, properties connected with the divisibility of the cycle's periods, the divisibility of the longest cycle period T(d) and the dimension d, the structure of ratios T(d)/d, the structure of the attached trees, and many other notions. We will introduce the definition of the complexity of integers and prove some properties of the associated graph by means of finite vector spaces.

Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by nevins.
 Prakash Belkale (University of North Carolina)The monodromy of the Hitchin/WZW connection, and applications to strange dualitiesAbstract: Non abelian theta functions, which generalise the classical theta functions on the Jacobian of a curve X, are associated to a group G, an algebraic curve X and a level k. As the curve X moves in its moduli space, the spaces of non abelian theta functions carry a connection. It is an open problem to determine the monodromy of this projective connection and to prove the physics expectation that it is unitary. I will review some of these questions. If G->H is a map of groups, H-theta functions map to G-theta functions. I will discuss the problem of determining when these maps are flat, with applications to strange dualities''.

Analytic Aspects of Quantum Groups
3:00 pm   in 130 Lincoln Hall,  Tuesday, October 21, 2008
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Submitted by pfima.
 Florin Boca (Department of Mathematics, University of Illinois)Unitary representations of compact quantum groups IIIAbstract: We will prove the Peter-Weyl Theorem for unitary representations of a compact quantum group

Graph Theory and Combinatorics
3:00 pm   in 241 Altgeld Hall,  Tuesday, October 21, 2008
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Submitted by west.
 Kevin Milans (Department of Mathematics, University of Illinois)Binary subtrees with few path labelsAbstract: A k-ary tree is a rooted tree in which all non-leaves have k children; it is complete and has depth n if all leaves have the same distance n from the root. Let T be the complete ternary tree of depth n. If each edge in T is labeled 0 or 1, then the labels along the edges of a path from the root to a leaf form a "path label" in &ob;0,1&cb;n. Let f(n) be the maximum, over all &ob;0,1&cb;-edge-labeled complete ternary trees with depth n, of the minimum number of distinct path labels on a complete binary subtree of depth n. The problem of bounding f(n) arose in studying a problem in computability theory, where it was hoped that f(n)/2n tends to 0 as n grows. This is true; we show that f(n)/2n is O(2-c√n) for some positive constant c. From below, we show that f(n)≥(1.548)n for sufficiently large n. This is joint work with Rod Downey, Noam Greenberg, and Carl Jockusch.