Department of

Mathematics


Seminar Calendar
for events the day of Thursday, January 20, 2005.

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Thursday, January 20, 2005

11:00 am in 241 Altgeld Hall,Thursday, January 20, 2005

Equivariant Elliptic Cohomology

Jacob Lurie (Harvard)

1:00 pm in 241 Altgeld Hall,Thursday, January 20, 2005

The correlations of Farey fractions

Florin Boca (UIUC)

Abstract: It will be proved that all correlation measures of the sequence of Farey fractions exist. The pair correlation function is explicitly computed and shows an intermediate behaviour between the Poisson and the GUE correlations. This is joint work with A. Zaharescu.

2:00 pm in 345 Altgeld Hall,Thursday, January 20, 2005

Geometry of Hermitian algebraic functions

Dror Varolin   [email] (UIUC Math)

Abstract: I will consider a class of functions, called Hermitian algebraic, whose domains are the total space of a holomorphic line bundle. I will define these functions, and show that they are generalizations of very natural mathematical objects, namely bihomogeneous Hermitian polynomials. I will talk about a few elementary properties of Hermitian algebraic functions, and then proceed to state a Hermitian algebraic analogue of Hilbert's 17th problem for such functions. (Recall Hilbert's 17th problem = Artin's Theorem: Any non-negative polynomial is a sum of squares of rational functions.) The analog can be vaguely stated as follows: Is any non-negative Hermitian algebraic function a quotient of squares of holomorphic mappings? This Hermitian algebraic analogue of Hilbert's 17th problem was considered by D. Quillen in a certain important case, and later by D'Angelo and Catlin in the general setting. Both Quillen and Catlin-D'Angelo settled the problem in the affirmative under the hypothesis that the Hermitian function is strictly positive away from the zero section. D'Angelo also showed that in the Hermitian algebraic setting, there are non-negative Hermitian algebraic functions that are not quotients of squares. I will state necessary and sufficient conditions under which a non-negative Hermitian algebraic function is a quotient of squares. I will then indicate how to prove this result in the case considered by D'Angelo-Catlin. If time permits, I will indicate how one goes on to the general case.

4:00 pm in 245 Altgeld Hall,Thursday, January 20, 2005

Stability of periodic solutions in dynamical phase transition models

Kevin Zumbrun (Indiana University)

Abstract: In this lecture, we discuss stability and asymptotic behavior of solutions of phase-transitional models arising in van der Waals gas dynamics and elasticity and in multiphase flow in porous media. Such models present a rich solution structure resembling that found in their cousin Cahn--Hilliard or Allen--Cahn models (reaction--diffusion models, vs. the convection--diffusion models we study), including pulse-type and spatially periodic traveling waves along with the more usual front-type (shock) solutions expected in compressible flow. Likewise, they exhibit behaviors of pattern formation, nucleation, etc. similar to that found in the reaction-diffusion setting. On the other hand, the equations retain the technical difficulty of compressible gas dynamics; in particular, we cannot determine stability by variational principles alone, as for the gradient flow models mentioned above. Instead, we make use of a periodic Evans function introduced recently by R.A. Gardner to study directly the stability of spatially periodic traveling waves. However, we use it in a different way than it was used by Gardner, exploring stability by geometric considerations rather than an asymptotic limit, in the spirit of the original work of J. Evans in the context of traveling front- or pulse-type solutions.