Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 8, 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, February 8, 2005

11:00 am in Altgeld Hall 241,Tuesday, February 8, 2005

#### Topological Hochschild cohomology of Morava K-theory

###### Vigleik Angeltveit (MIT)

Abstract: I will define Hochschild cohomology and topological Hochschild cohomology as derived functors of the center, and give some examples. Baker and Lazarev have shown that THH(KU/2)=KU_2, 2-completed K-theory. I will explain this result, and how it continues to hold at odd primes, if we equip KU/p with a non-commutative multiplication. If we instead consider THH(K(1)), I will try to argue that we get E(1)^[a]/(p=a^(p-1)v_1), i.e., the completed Johnson-Wilson spectrum adjoined a (p-1)st root of pv_1^-1, though this is still work in progress.

1:00 pm in Altgeld Hall,Tuesday, February 8, 2005

#### Simplicity of Pseudofinite Fields.

###### Dominika Polkowska (UIUC Math)

Abstract: In this talk a detailed outline of the proof of simplicity of pseudofinite fields will be given. In fact, the proof I will present works in much more general setting i.e. for bounded PAC substructures of stable structures. This is one of the main results on my thesis.

1:00 pm in 241 Altgeld Hall,Tuesday, February 8, 2005

#### Effective structure theorems for quadratic spaces and their isometries

###### Lenny Fukshansky, (Texas A&M)

Abstract: A classical theorem of Witt states that a bilinear space can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. I will discuss the existence of such a decomposition of bounded height for a symmetric bilinear space over a number field, where all bounds on height are explicit. I will also talk about an effective version of Cartan-Dieudonné theorem on representation of an isometry of a regular symmetric bilinear space as a product of reflections. Finally, if time permits, I will show a special version of Siegel's Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces.

2:00 pm in 243 Altgeld Hall,Tuesday, February 8, 2005

#### Geometry problems from the Putnam exam

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

Abstract: Solutions will be presented to three problems posed on the recent Putnam exam. (1) If one triangle has all three sides at least as long as a second one, is the area of the first greater than that of the second? (2) For what lengths of interval of definition does the region below some graph of a continuous positive function y = f(x) have area equal to the perimeter? (3) If n plane rotations each have angle 360/n degrees and their centers are equally spaced along a line, what is the product of these rotations in the same order as the linear order of the centers?

2:00 pm in Siebel Center 3401,Tuesday, February 8, 2005

#### Bovine Harmonics

###### P.-T. Bremer (UIUC Computer Science)

2:00 pm in 345 Altgeld Hall,Tuesday, February 8, 2005

#### Hamiltonian systems and generation of vortex structures in fluid motion

###### Mikhail Fokin (Sobolev Institute of Mathematics, Novosibirsk, Russia)

Abstract: The purpose or this talk is to demonstrate new analytical method for studying of the generation of vortex structures in a class of three dimensional incompressible fluid flows. These flows are invariant under the action of a one-parameter symmetry group. It followes in this case that there exists the coordinate system such that the evolution of two of the coordinates is governed by a time-dependent Hamyltonian system with the evolution of remaining coordinate being governed by a first order differential equation that depends only on the other two coordinates and time. The moving vortex structures in the fluid which correspond to local maxima and minima of the Hamiltonian function for given value of time are similar to a tornado. The process of arising, evolution and disappearance of the vortex structures is described. The problem of small oscillations of rotating ideal fluid is considered as the example. The characteristics of spectra in this problem were investigated in details in the case when the velocity and the pressure are dependent only of two spatial variables and the fluid domain is an endless cylinder with the convex domain in the base. The structure of infinite-dimensional manifolds of initial disturbances of the velocity, which correspond to the various intervals of the energy spectrum, is described. The typical phase portraits of dynamic systems which characterize the oscillations of fluid particles are obtained for each manifold of initial data. The number of vortex structures increases in time and their scale decreases for the motions with continuous energy spectrum. This effect may be considered as one of the mathematical models of development of turbulence. The specific features of this evolution, which correspond to singular continuous spectrum, are characterized.

3:00 pm in 347 Altgeld Hall,Tuesday, February 8, 2005

#### Free Central limit Theorem

###### Mingchu Gao (UIUC)

Abstract: We will show some examples of the computation of R-transforms such as semicircle laws. We will prove the freely central limit theorem. We also plan to talk on random walks on free groups if we have extra time.

3:00 pm in 241 Altgeld Hall,Tuesday, February 8, 2005

#### Generalizations of the Erdos-Ko-Rado Theorem

###### Kyung-Won Hwang (UIUC Math)

Abstract: In 1961, P. Erdos, C. Ko, and R. Rado proved that if F is a k-uniform family of subsets of an n-set, with k <= n/2, and every two members of F intersect, then |F| <= {n-1 \choose k-1}. Many people generalized this theorem with one size k. We generalize the Erdos-Ko-Rado Theorem allowing several sizes ki. This is joint work with Zoltan Furedi and Paul M. Weichsel.

4:00 pm in 341 Altgeld Hall,Tuesday, February 8, 2005