Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, November 1, 2005.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, November 1, 2005

1:00 pm in 345 Altgeld Hall,Tuesday, November 1, 2005

#### The Product Rule for Derivations on Lie Algebras

###### Richard Kramer (Iowa State)

1:00 pm in 347 Altgeld Hall,Tuesday, November 1, 2005

#### Do there exit homogeneous Calderon-Zygmund singular integrals bounded on some, but not all, L^p spaces

###### Loukas Grafakos   [email] (University of Missouri-Columbia, Math)

2:00 pm in 243 Altgeld Hall,Tuesday, November 1, 2005

#### Caustics of Circular Billiards

###### Prof. Richard L. Bishop (UIUC Department of Mathematics)

Abstract: The qualitative and quantitative properties of the caustics of a circular mirror are described. For a point source, the nth caustic is the envelope of the family of lines containing the (n+1)st reflective segment. It is an analytic curve with four cusps and may extend outside the circle, depending on n and the distance of the source from the center. If we view the paths as geodesics of a double disk, then the nth conjugate locus of the source consists of arcs of one or two of the caustics. The simplest of these is the first conjugate and caustic locus of a point on the bounding circle, which was already identified by Huygens as a cardioid.

2:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2005

#### Purification of Measure-Valued Maps, III

###### Peter Loeb (UIUC Math)

Abstract: In 1951, Dvoretzky, Wald and Wolfowitz used the Lyapunov theorem for vector measures to establish a result on the purification of measure-valued maps from a nonatomic probability space T to a finite action space A. This theorem justifies the elimination of randomness in various settings such as the game theory of Nash equilibria. In this talk, which is joint work with Yeneng Sun, we generalize the result to a complete separable metric space A. Even when A is a closed, finite interval in the real line, however, an example shows that the extension fails when T is the unit interval supplied with Lebesgue measure and another measure having a continuous density function. To obtain our extension, we require that T with its associated measures are nonatomic measure spaces of the kind introduced by the speaker, and now called "Loeb spaces" in the literature.

2:00 pm in 152 Henry,Tuesday, November 1, 2005

#### No meeting this week

3:00 pm in 243 Altgeld Hall,Tuesday, November 1, 2005

#### Lie theory for differential graded Lie algebras

###### Ezra Getzler   [email] (Northwestern University)

Abstract: We present a nonabelian version of the Dold-Kan correspondence. Our correspondence associates to a dg Lie algebra concentrated in degrees (-n,0] a Kan complex which we identify as the nerve of an n-groupoid.

3:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2005

#### On a graph packing conjecture

###### Gexin Yu (UIUC Math)

Abstract: Two n-vertex graphs G1 and G2 pack if G1 and G2 can be expressed as edge-disjoint subgraphs of Kn. Special cases of the problem of whether two given graphs pack include problems on existence of fixed subgraphs, on forbidden subgraphs, and on equitable coloring. Graph packing has also been applied in studying computational complexity of graph properties. Let \Delta(G) denote the maximum degree of a vertex in G.

Bollobás and Eldridge (also Catlin, independently) conjectured that if |V(G1)|=|V(G2)|=n and (\Delta(G1)+1)(\Delta(G2)+1) <= n+1, then G1 and G2 pack. If true, this conjecture would be sharp, and it would be a considerable extension of the Hájnal-Szemerédi theorem on equitable coloring. The conjecture has only been proved in cases where G1 is highly degenerate, or \Delta1 <= 2, or \Delta1=3 and n is huge.

In this talk, we take a different approach to the conjecture. Given n-vertex graphs G1 and G2, we prove that if \Delta(G1), \Delta(G2) >= 400 and (\Delta(G1)+1)(\Delta(G2)+1) <= 0.6n+1, then G1 and G2 pack. This is joint work with H. Kaul and A. Kostochka.

4:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2005

#### Types and Ranks in Differentially Closed Fields

###### Sonat Suer (UIUC Math)

Abstract: We will give an algebraic characterization of forking in DCF_0 for 1-types and give two "bad" examples in which certain notions of dimension do not agree