Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 28, 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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30


Tuesday, March 28, 2006

1:00 pm in 243Altgeld Hall,Tuesday, March 28, 2006

#### The Birational Geometry of Moduli Spaces of Stable Curves

###### Professor Ian Morrison (Fordham University dept of math)

Abstract: I will review what we know about the birational geometry of moduli spaces of stable curves, beginning with the origins of the subject in the 19th century (dimension, irreducibility, ...) and ending with current work and open problems (ample and effective cones, structure of the canonical ring, ...) and, if time permits, a few applications. The talk will be expository and assume only a basic familiarity with moduli spaces or birational geometry.

1:00 pm in 241 Altgeld Hall,Tuesday, March 28, 2006

#### Report from the "Anatomy of Integers" workshop in Montreal

###### Kevin Ford (UIUC)

1:00 pm in 345 Altgeld Hall,Tuesday, March 28, 2006

#### Extreme amenability of L0

###### Slawomir Solecki (UIUC Math)

Abstract: I will show that L0(φ, S1) is extremely amenable for any diffused submeasure φ. This extends results of Herer--Christensen (for φ a submeasure with no non-zero measure below) and Glasner and Furstenberg--Weiss (for φ a measure). Proofs of these earlier results used spectral theory or concentration of measure. Our argument is based on a Ramsey-type theorem proved using ideas coming from combinatorial applications of algebraic topological methods. This is based on parts of joint work with I. Farah.

1:00 pm in 347 Altgeld Hall,Tuesday, March 28, 2006

#### Birman-Schwinger Principle

###### Richard S. Laugesen (UIUC Math)

Abstract: We will explain this famous principle, which estimates the number of negative bound states of a Schrodinger operator in terms of the number of eigenvalues > 1 of a related integral operator.

2:00 pm in 243 Altgeld Hall,Tuesday, March 28, 2006

#### On k-sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn.

###### Prof. Jozsef Balogh (UIUC Department of Mathematics)

Abstract: We use circular sequences to give an improved lower bound on the minimum number of (< k)-sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number of convex quadrilaterals determined by the points in S is at least 0.37553(n choose 4) + O(n3). This in turn implies that the rectilinear crossing number cr(Kn) of the complete graph Kn is at least 0.37553(n choose 4) + O(n3), and that Sylvester's Four Point Problem Constant is at least 0.37553. These improved bounds refine results recently obtained by Ábrego and Fernández--Merchant, and by Lovász, Vesztergombi, Wagner and Welzl.

This reports joint work with Gelasio Salazar.

2:00 pm in 347 Altgeld Hall,Tuesday, March 28, 2006

#### Stochastic porous media and fast diffusion equations

###### Michael Roeckner (Purdue University)

Abstract: We first present a new existence and uniqueness result for stochastic evolution equations on Hilbert spaces. This is a generalization of a classical result by Krylov and Rozovskii based on the so-called variational approach to stochastic partial differential equations (SPDE). The main motivation are applications to nonlinear SPDE of porous media type which also include cases where the nonlinear functions grow slowly at infinity ("fast diffusion equations"). Generally, the main problem is to find the appropriate Gelfand triple to work on. In our case Orlics spaces turn out to be convenient. We show how one must choose the defining Young function for a given nonlinearity. After presenting these applications, we shall summarize results about the qualitative behaviour of solutions and about their invariant measures.

3:00 pm in 443 Altgeld Hall,Tuesday, March 28, 2006

#### Nonstandard representations of Young measures, cont

###### 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.

3:00 pm in 245 Altgeld Hall,Tuesday, March 28, 2006

#### An Ore-type analogue of the Sauer-Spencer Theorem

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

Abstract: Graphs G1 and G2 with the same number (n) of vertices pack if there exist injective mappings of their vertex sets into [n] such that the images of the edge sets do not intersect. Sauer and Spencer proved that if \Delta(G1) \Delta(G2) < n/2, then G1 and G2 pack. In this talk, we investigate an Ore-type analogue of the Sauer-Spencer Theorem. Let \theta(G)=\max{d(u)+d(v): uv\in E(G)}. We show that if \theta(G1)\Delta(G2) < n, then G1 and G2 pack. We also characterize the pairs (G1,G2) of n-vertex graphs that do not pack but satisfy \theta(G1)\Delta(G2)=n. Some open problems will be mentioned as well. This is joint work with Prof. Kostochka.

4:00 pm in 245 Altgeld Hall,Tuesday, March 28, 2006

#### A Mathematician Returns To Elementary School

###### Prof. John D'Angelo (UIUC Math)

Abstract: Prof. D'Angelo will share some of his experiences as an occasional volunteer teacher at Leel Elementary School. He will then compare these experiences with teaching U of I students, especially those in Math 347, a transition course for math majors

5:00 pm in 343 Altgeld Hall,Tuesday, March 28, 2006

Abstract: TBA