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Seminar Calendar
for events the day of Tuesday, January 31, 2006.

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Tuesday, January 31, 2006

1:00 pm in 441 Altgeld Hall,Tuesday, January 31, 2006

Complexes and Betti numbers

Bart Snapp (UIUC Math)

Abstract: We'll discuss simplicial complexes and homology, Koszul complexes, Betti numbers, and a notion of duality.

1:00 pm in 345 Altgeld Hall,Tuesday, January 31, 2006

Local stability in the setting of continuous first order logic.

Sylvia Carlisle (UIUC Math)

Abstract: In "Continuous first order logic and local stability" Itay Ben-Yaacov and Alexander Usvyatsov define local stability for the continuous first order logic setting and present results characterizing the stability of a formula analogous to well known results from first order logic. I will give an introduction to continuous first order logic, contrasting it with usual first order logic. I will also present a proof that, as in first order logic, the stability of a formula phi is equivalent to a restriction on the size of the space of phi-types.

2:00 pm in 243 Altgeld Hall,Tuesday, January 31, 2006

Borsuk-Ulam implies Brouwer--a direct construction

Mary-Elizabeth Hamstrom (UIUC Department of Mathematics)

Abstract: The Brouwer fixed point theorem states that if f is a continuous map of the n-ball Bn to itself, then there exists x such that f(x) = x. One usually proves this by saying "suppose not" and then appealing to degree or homology to get a contradiction. The machinery gives a proof of Borsuk-Ulam: if f maps Sn to En there is an x such that f(-x) = f(x) (i.e., some pair of antipodal points goes to the same point).

Francis Edward Su (Amer. Math. Monthly 104 (1997) p.855-859) gives an elementary geometric direct construction proving that Borsuk-Ulam implies Brouwer. Let f be a map of Bn to itself. We construct a map g of the n-sphere Sn into En, Euclidean n-space (which contains the n-ball). An x such that g(-x) = g(x) will give us a fixed point of f.

Some major machinery is needed to get Borsuk-Ulam directly, so this is not an elementary proof of Brouwer, a fact that the author does not make clear. However, the construction is nice and requires only the knowledge of n-dimensional geometry to be found in an elementary linear algebra course or a good calculus course.

3:00 pm in 241 Altgeld Hall,Tuesday, January 31, 2006

Parity edge-coloring of graphs

Douglas B. West (UIUC Math)

Abstract: A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G with no parity path (called a parity edge-coloring. Let p'(G) be the least number of colors in an edge-coloring of G in which every parity walk is closed (called a strong parity edge-coloring). Always p'(G) >= p(G) >= chi'(G).

A graph G embeds in the k-dimensional hypercube if and only if p(G) <= k and every cycle in G is a parity walk. Consequently, p(G) >= ceiling(lg n(G)), and equality holds for paths and even cycles. When n is odd, p(Cn)=p'(Cn)=1+ceiling(lg n). Also, p(K2,n)=p'(K2,n), with value n when n is even and n+1 when n is odd. The main result is that p'(Kn)=2ceiling(lg n)-1 for all n. Furthermore, the optimal coloring for Kn is unique when n is a power of 2 and completely described for all n. Also p(Kn)=p'(Kn) when n <= 16.

3:00 pm in 443 Altgeld Hall,Tuesday, January 31, 2006

A description of sofic groups in terms of nonstandard analysis, cont.

Evgeny Gordon (Eastern Illinois University)

Abstract: NOTE THE CHANGE IN TIME AND LOCATION FOR THIS SEMINAR. Sofic groups were first defined by M. Gromov as a common generalization of amenable groups and residually finite groups. B. Weiss proved that the old problem of Gottschalk on surjunctive groups has a positive solution for sofic groups. The problem of existence of non-sofic groups remains open. In this talk we discuss this topic in terms of nonstandard analysis, which allows us to simplify some definitions and proofs.

4:00 pm in 245 Altgeld Hall,Tuesday, January 31, 2006

Elliptic curves, quadratic twists and L-values

Kartik A. Prasanna (UCLA)

Abstract: The first half of the talk will be an elementary discussion of elliptic curves, their associated L-functions and the structure of the Birch and Swinnerton-Dyer conjecture (BSD) for the behaviour of such L-functions at the central point s=1. In the second half, I will begin by explaining a theorem of Waldspurger on nonvanishing of central values of quadratic twists, and its applications, most notably to BSD. Finally, I will formulate a conjectural "mod p" analog of Waldspurger's theorem and describe some recent results that are related to this problem. Host: B. Berndt