UIUC Dept. of Mathematics Seminar Calendar

    September 2009          October 2009          November 2009    
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Tuesday, October 13, 2009

Topology seminar
11:00 am in 241 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by bertg.

Jeremiah Heller (Northwestern University)
Vanishing Theorems for Real Algebraic Cycles
Abstract: Homotopy groups of topological spaces of cycles on an algebraic variety form rich and intriguing invariants of the variety. Some of these homotopy groups are classical topological invariants (e.g. singular homology in the complex case, Bredon homology in the real case) and some are classical geometric invariants (e.g. cycles modulo algebraic equivalence). However most of these homotopy groups remain quite mysterious and difficult to compute. We discuss recent joint work with M. Voineagu where we show that the k-th homotopy group of the space of "reduced" r-cycles on a real variety vanishes for k larger than dim(X)-r. This is the real analogue of the (still open) Friedlander-Mazur conjecture for homotopy groups of cycles on a complex variety. Additionally we compute equivariant homotopy groups of spaces of cycles on a real variety in terms of Bredon homology in some cases.

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by verahur.

Francisco Gancebo (University of Chicago)
Contour dynamics for 2D active scalars
Abstract: In this talk we discuss two free boundary problems given by fluid domains which are weak solutions of incompressible equations. We consider the contour dynamics Muskat problem and the evolution of a sharp front by the 2D surface Quasi-geostrophic equation. Both systems are described by means of a transport equation for the active scalar \rho(x,t) which takes constant values on complementary domains. The velocity field is determined by \rho(x,t) by singular integral operators. However the solutions of these two physical scenarios have completely different outcomes regarding well-posedness and regularity issues.

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by w-henson.

Yevgeniy Gordon (EIU)
What does G. Birkhoff's Ergodic Theorem mean for very big finite spaces? (part I)
Abstract: The trivial proof of the ergodic theorem for a finite set X and a permutation s of X shows that for an arbitrary real-valued function f on X, the sequence of ergodic means A_n(f,s) stabilizes for large n. We show that if X is a very big finite set and f(x) is much less than the size of X for almost all x, then A_n(f,s) stabilizes for a significantly long segment of big numbers n that are, however, much less than the size of X. This statement has a natural rigorous formulation in terms of nonstandard analysis, which is, in fact, equivalent to the ergodic theorem for infinite probability spaces. Its standard formulation in terms of sequences of finite probability spaces is complicated. We discuss also a new notion of approximation of dynamical systems by finite dynamical systems, whose definition in terms of nonstandard analysis is much easier and much more natural than in classical terms. This is joint work with Lev Glebsky and Ward Henson.

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by llpku.

Wenchuan Hu (Institute of Advanced Study)
Topological aspects on Chow varieties
Abstract: The topological invariants of Chow varieties can be calculated by the homotopy theoretic method. In this talk I will discuss this method in calculating the Euler Characteristic of Chow varieties. This result has been obtained by Blaine Lawson and Stephen Yau by using a fixed point formula with a torus action. Our calculation in a direct and simple way. This technique also can be generalized to Chow varieties with certain group actions and other cases. Furthermore, I will also talk about the application of the method on l-adic Euler-Poincare Characteristic of Chow varieties over arbitrary algebraic closed field.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by west.

Dominic Searles (UIUC Math)
Comparative probability orders and the flip relation
Abstract: A comparative probability order is a linear order on the subsets of a finite set X, starting with ∅, such that whenever A precedes B and C∩(A∪B) = ∅, also A∪C precedes B∪C. These orders have been used to study betting behaviour, where an agent believes some events are more likely than others and bets accordingly.

Nonempty disjoint sets A and B that are consecutive in a c.p.o are flippable if for every C such that C∩(A∪B)=∅, the sets A∪C and B∪C are also consecutive. Exchanging the two sets in a flippable pair (and the pairs A∪C and B∪C such that C∩(A∪B)=∅) yields an "adjacent" comparative probability order. We will describe representations of comparative probability orders such as discrete cones and geometric polytopes, explain what the flip relation represents in these contexts, and discuss a conjecture about the maximum number of neighbours (under flipping) a comparative probability order may have.