Mathematics

Sums of roots of unity

Alexandru Zaharescu (UIUC)

The Hairy Ball Theorem via Sperner's Lemma

Prof. Mary-Elizabeth Hamstrom (UIUC Department of Mathematics)

Abstract: I report on a paper of that title by Tyler Jarvis and James Tanton in the American Mathematical Monthly (111 (2004) 599-603). This all takes place in dimension 2. A slight generalization of Sperner's Lemma is proved: If K is a triangulation of a polygonal disc whose vertices are labeled A,B,C so that at least one boundary edge is labeled AB and all AB edges on the boundary are oriented in the same way, then there is a triangle of K whose vertices are labeled ABC. The proof is entertaining and elementary.

This is used to prove the Hairy Ball Theorem: There is no continuous non-vanishing tangent vector field on the 2-sphere. In plainer language, one cannot assign to each point x of S a unit vector T(x) tangent to S at x, T(x) varying continuously with x. This theorem is true for all even dimensional spheres, but its proof requires the powerful machinery I mentioned in my last talk. For odd dimensional spheres such a tangent vector field always exists. All this is elementary.

Quasi-invariance of the Wiener measure in loop spaces

Elton Hsu (Northwestern University)

Abstract: A Cameron-Martin path generates geometrically a vector field on the loop space over a compact Riemannian manifold. Following the works of Driver and Enchev/Stroock, we discuss some new ideas in proving that the Wiener measure (Brownian bridge) is quasi-invariant under the flow generated by this vector field.

The circular chromatic index of graphs of high girth

Daniel Král (Georgia Tech and Charles University (Prague))

Abstract: Colorings of graphs form a prominent topic in graph theory. Several relaxations of ordinary colorings have been introduced and intensively studied. In this talk, we focus on circular colorings of line graphs. A proper circular k-edge-coloring, for a real k >= 1, is a coloring by real numbers from the interval [0,k) such that the difference modulo k of the colors c₁ and c₂ assigned to incident edges is at least 1; that is, 1 >= |c₁-c₂| >= k-1.

A classical theorem of Vizing states that the edges of every graph G with maximum degree D can be colored by at most D+1 colors so that no two incident edges have the same color; that is, the chromatic index of G is at most D+1. We show that for every e>0 there exists g such that the circular chromatic index of a graph G with maximum degree D whose girth is at least g does not exceed D+e. Note that the index must be at least D because the line graph of such a graph contains a clique of order D.

Our research is motivated by a conjecture of Jaeger and Swart 1979 (which turned out to be false) that high girth cubic graphs have chromatic index 3. Our results imply that the relaxation of the conjecture to circular colorings is true: the circular chromatic index of high girth cubic graphs is close to 3. Among the ingredients of our proof are recent results on systems of independent representatives and hypergraph matchability by Aharoni, Haxell, Meshulam, and others, which we also briefly survey during the talk. This work is joint with Tomas Kaiser, Riste Skrekovski, and Xuding Zhu.

Nonstandard representations of Young measures

David Ross (visiting UIUC from Hawaii 05-06)

Abstract: Young measures are measure-valued functions that arise as generalized limits of sequences of measurable functions with increasing oscillation. Nonstandard analysis allows for natural representations for both Young measures and the usual (narrow) topology on the space of Young measures.