Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 22, 2005.

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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 22, 2005

1:00 pm in 464 Loomis,Tuesday, February 22, 2005

#### Quivers for Metrics

###### Prof Amihay Hanany (MIT Physics)

1:00 pm in 345 Altgeld Hall,Tuesday, February 22, 2005

#### Haar null sets and local amenability

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

Abstract: I will talk about a class of Polish groups whose definition is obtained by suitably localizing to the identity element of the notion of amenability. I will present results showing that on such groups left Haar null sets (measure theoretically small sets) have many desired properties (they form a $\sigma$-ideal and have the Steinhaus property). I will also outline the extent of this class of Polish groups. It turns out that opposite to Polish groups which are amenable at the identity are Polish groups which have a non-Abelian free subgroup at the identity. I will make this precise. For such groups left Haar null sets lose the essential properties they enjoy for amenable at $1$ groups. I will present these results as well.

1:00 pm in 241 Altgeld Hall,Tuesday, February 22, 2005

#### An Extension of the Supercongruence for Apery Numbers

###### Timothy Kilbourn (UIUC)

Abstract: In 1987, Beukers proved a mod p congruence between the coefficients of a certain modular form and the Apery numbers. He conjectured the existence of a supercongruence'' mod p^2; this was proved by Ahlgren and Ono in 2000. In this talk we prove an extension of this congruence mod p^3. The proof involves the modularity of a Calabi-Yau threefold, the Gross-Koblitz formula, the p-adic gamma function, and some interesting combinatorics.

2:00 pm in 241 Altgeld Hall,Tuesday, February 22, 2005

#### Henn's calculation of the cohomology of S_2 at p=3, III

###### Charles Rezk (UIUC Math)

2:00 pm in 243 Altgeld Hall,Tuesday, February 22, 2005

#### A new collinearity

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

Abstract: I will discuss a recent article by Apostal and Mnatsakanian ("Figures circumscribing circles," Amer. Math. Monthly 111 (2004) 853-863). If O is the incenter of a triangle, C(A) is the vector from O to the area centroid A (intersection of the medians) and C(B) is the vector from O to the boundary centroid B, then O, C(A), and C(B) are collinear and C(B)=(3/2)C(A). The authors prove this for a large class of polygons P, not necessarily simple or closed, whose edgelines are tangent to a circle, called the incircle of P, and in particular for all polygons that circumscribe a circle. The talk will be quite elementary.

3:00 pm in 243 Altgeld Hall,Tuesday, February 22, 2005

#### Introduction to Frobenius Splitting

###### William Haboush (UIUC)

Abstract: Abstract: I will define Frobenius split varieties. I will then review finite duality theory and the relative canonical bundle of the Frobenius morphism. I will explain the Borel Weil Theorem on a generalized flag variety and attempt to show that it is a Frobenius split variety.

3:00 pm in 241 Altgeld Hall,Tuesday, February 22, 2005

#### A note on semiantichains and unichain coverings

###### Qi Liu (UIUC Math)

Abstract: Saks and West conjectured that for every direct product of partial orders, the maximum size of a semiantichain equals the minimum number of unichains needed to cover the product. In this talk we will prove that when both posets have width 2, the conjecture is true. This is joint work with Douglas West.

3:00 pm in 347 Altgeld Hall,Tuesday, February 22, 2005

#### Asymptotic Freeness

###### Mingchu Gao (UIUC)

Abstract: We will define the limit distribution and asymptotic freeness of a family of random variables. Then we will prove a result on sufficient conditions of asymptotic freeness.

4:00 pm in 341 Altgeld Hall,Tuesday, February 22, 2005

#### Extending partial automorphisms of finite structures: short extensions

###### Ayhan Gunaydin (UIUC Math)

4:00 pm in 245 Altgeld Hall,Tuesday, February 22, 2005

#### Modeling Large Societies

###### Yeneng Sun (National University of Singapore)

Abstract: There are many diverse examples of groups with a large number of competing participants. As early as 1944, in their famous book "Theory of Games and Economic Behavior", von Neumann and Morgenstern emphasized the importance of studying mass phenomena arising from the interaction of a large number of participants. To model the idea that the influence of individuals is negligible in very large societies, Milnor and Shapley in 1961 used a non-atomic measure space to represent "a continuum of infinitesimal minor players." Since then, many new results have been obtained in the continuum setting that would fail in the case of a fixed finite number of agents. It has been discovered recently, however, that the usual continuum model such as Lebesgue space may not appropriately capture limiting phenomena for the interaction of a large, but finite, number of agents. Examples include the disappearance of Nash equilibria for games with many players, and the lack of a limiting model for many agents with independent risks. This talk will show how rich measure spaces, such as Loeb measure spaces, can be used not only to resolve these problems, but also to go beyond them to discover unexpected new connections in economics as well as in mathematics.