Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 7, 2006.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, February 7, 2006

1:00 pm in 241 Altgeld Hall,Tuesday, February 7, 2006

#### Congruences and Ideals

###### Heini Halberstam (UIUC Math )

1:00 pm in 441 Altgeld Hall,Tuesday, February 7, 2006

#### Complexes and Betti numbers II

###### Bart Snapp (UIUC Math)

Abstract: This is a continuation of last week's talk. We'll discuss simplicial complexes and homology, Koszul complexes, Betti numbers, and a notion of duality.

1:00 pm in 345 Altgeld Hall,Tuesday, February 7, 2006

#### A characterization of locally compact subgroups of S∞

###### Maciej Malicki (UIUC Math)

Abstract: It is known that a Polish group, G, is compact iff for every open set, U, there exists a finite covering of G by two sided translates of U. This result is pretty obvious for abelian groups and, more generally, groups that admit an invariant compatible metric. However, there are many interesting Polish groups that do not satisfy this. I will prove a counterpart of the above characterization for locally compact subgroups of S, the group of permutations of the natural numbers, and give examples of some other groups that satisfy it. Also, I will show how these results allow one to strengthen some theorems on Haar null sets.

2:00 pm in 243 Altgeld Hall,Tuesday, February 7, 2006

#### Simulating perturbation to enumerate parametrized systems

###### Dr. Daniel Lichtblau (Wolfram Research, Inc.)

Abstract: A problem frequently encountered in geometric constraint solving and related settings is to ascertain the number of solutions generically expected from a well constrained input configuration, e.g. counting lines simultaneously tangent to four given spheres. Once translated to an algebraic setting one has a system of polynomial equations with some coefficients parametrized, and wants to determine how many solutions it has. Typically this will be guessed by resorting to random coefficients, and perhaps rigorously proved by other means. We describe a simple and practical algorithm and implementation using Groebner bases that will provide the correct generic solution count to such systems. We demonstrate on several examples, recovering some known but nontrivial results and confirming a recent conjecture concerning metric invariants of tetrahedra.

I intend to keep the technicalities at bay, so this talk will be accessible to a wide audience.

3:00 pm in 443 Altgeld Hall,Tuesday, February 7, 2006

#### Two talks

###### Evgeny Gordon (1st) and David Ross (2nd)

Abstract: First, Evgeny Gordon will finish proving that the "universal sofic group" from nonstandard analysis is simple. Then David Ross will take over: Subject: Pushing down infinite Loeb measures Description: Sufficient conditions are given under which the standard part map on a locally compact Hausdorff space can be used to push down an infinite nonstandard measure. This makes it easier to construct standard infinite Borel measures using nonstandard techniques.

3:00 pm in 241 Altgeld Hall,Tuesday, February 7, 2006

#### Recent advances in domination on graphs

###### Michael D. Plummer (Vanderbilt University)

Abstract: A set D of vertices in a graph G is a dominating set if every vertex of G not in D has a neighbor in D. The size of a smallest dominating set, denoted \gamma(G), is the domination number of G. We report on recent results with N. Ananchuen, with X. Zha, and with K. Kawarabayashi and A. Saito about three different topics involving domination in graphs.

A graph G is \gamma-edge-critical if \gamma(G+e)<\gamma(G) for each edge e\notin E(G). It is \gamma-vertex-critical if \gamma (G-v)<\gamma(G) for every vertex v\in V(G). The structure of \gamma-edge-critical graphs and \gamma-vertex-critical graphs is not well understood, even when \gamma(G)=3. We present new results on both classes that involve matchings.

In 1996, Matheson and Tarjan proved that a triangulated disc with n vertices has domination number at most n/3, and thus so does every n-vertex triangulation. We will present recent work toward extending this result to graphs of higher genus.

Reed conjectured in 1996 that if G is a cubic graph with n vertices, then \gamma (G) <= \lceil|V(G)|/3\rceil. This conjecture was very recently shown to be false by Kostochka and Stodolsky. We will close by discussing new results pertaining to this conjecture.

5:00 pm in 343 Altgeld Hall,Tuesday, February 7, 2006

#### Distinguishing Chromatic Number of a graph

###### Jeong Ok Choi (UIUC Math)

Abstract: The distinguishing chromatic number of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We will show that the distinguishing chromatic number of G^d (the Cartesian product of G by d times) is at most one more than the usual chromatic number of G for d at least 6, where G is either a complete graph or a hypercube. In fact, with a larger value of d, the generalization for any (connected) graph is true. This is joint work with Hemanshu Kaul and Stephen Hartke.