Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, January 28, 2014.

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    December 2013           January 2014          February 2014    
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Tuesday, January 28, 2014

2:00 pm in 245 Altgeld Hall,Tuesday, January 28, 2014

Character rigidity in higher-rank groups

Jesse D. Peterson (Vanderbilt University)

Abstract: A character on a group is a class function of positive type. For finite groups, the classification of characters is closely related to the representation theory of the group and plays a key role in the classification of finite simple groups. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank Lie groups the space of characters should be completely determined by their finite dimensional representations. In this talk, I will give an introduction to this conjecture, and I will discuss its relationship to ergodic theory, invariant random subgroups, and von Neumann algebras.

3:00 pm in 241 Altgeld Hall,Tuesday, January 28, 2014

On (4m:2m)-choosability and a conjecture of Voigt

Gregory J. Puleo   [email] (UIUC Math)

Abstract: When G is a graph, L is a list assignment on G, and b is a nonnegative integer, we say that G is (L:b)-colorable if one can assign each vertex v a set of b colors from L(v) such that adjacent vertices receive disjoint sets. A graph is (a:b)-choosable if it is (L:b)-choosable whenever $|L(v)| \ge a$ for all vertices v; note that (k:1)-choosability is the same as ordinary k-choosability. This notion was introduced by Erdős, Rubin and Taylor in their first paper concerning choosability. It is known that all 2-choosable graphs are (2m:m)-choosable for all m, but the converse does not hold; in fact, it was shown by Alon, Tuza, and Voigt that every bipartite graph is (2m:m)-choosable for some (extremely large) value of m. A graph is 3-choosable-critical if its choice number is 3 and all its proper subgraphs are 2-choosable. Voigt showed that for odd m, G is (2m:m)-choosable if and only if G is 2-choosable, and conjectured that for even m, every bipartite 3-choosable-critical graph is (2m:m)-choosable. Resolving Voigt's conjecture in the negative, we determine which 3-choosable-critical graphs are (4:2)-choosable, and we conjecture a full characterization of (4:2)-choosable graphs. We also recover a weaker version of Voigt's conjecture, which we will discuss if time permits. This is joint work with Jixian Meng and Xuding Zhu.

4:00 pm in 245 Altgeld Hall,Tuesday, January 28, 2014

Integral points of bounded degree on curves

Aaron Levin (Michigan State University)

Abstract: A fundamental problem in number theory is to determine the set of solutions in integers to a system of polynomial equations. When the system of equations geometrically defines an affine curve, there is a fundamental result due to Siegel. Siegel's theorem asserts that an affine curve over a number field has only finitely many integral points if either the curve has positive genus or the curve has more than two points at infinity. Using powerful arithmetic results on points on abelian and semiabelian varieties, we generalize Siegel's theorem to integral points of degree d, giving a complete characterization of affine curves with infinitely many integral points of degree d (over some number field).