Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 17, 2005.

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Thursday, February 17, 2005

1:00 pm in 241 Altgeld Hall,Thursday, February 17, 2005

Congruences for the coefficients of weakly holomorphic modular forms, II

Stephanie Treneer (UIUC)

Abstract: Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomena is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form. In particular, we give congruences for a wide class of partitions functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. Tuesday's talk will consist of an introduction to the problem, a statement of the main theorems, and a discussion of the two applications. Thursday we will prove the main theorems.

1:00 pm in Altgeld Hall 347,Thursday, February 17, 2005

The intersection form and geodesic currents on free groups

Ilya Kapovich (UIUC)

Abstract: The notion of a geometric intersection number between free homotopy classes of closed curves on surfaces plays a pivital role in Thurston's treatment of the Teichmuller space and of the dynamics of surface homeomorphisms. In particular, Bonahon proved that this notion extends to a symmetric and bilinear notion of intersection number between two geodesic currents on a hyperbolic surface. We investigate to what extend these ideas are applicable in the free group context. Thus we define and study an Out(F_n)-equivariant "intersection form" on the product of the (non-projectivized) Culler-Vogtmann outer space and the space of geodesic currents on a free group. We also find an obstruction, arising from non-symmetric behaviour of generic stretching factors of free group automorphisms, to the existence of a symmetric notion of an intersection number between two geodesic currents on a free group.

2:00 pm in Altgeld Hall 243,Thursday, February 17, 2005

Testing analyticity on circles

Alex Tumanov (UIUC)

Abstract: Consider a continuous one parameter family of circles in complex plane that contains two circles lying in the exterior of one another. Under mild assumptions on the family, we prove that if a continuous function on the union of the circles extends holomorphically into each circle, then the function is holomorphic. This result partially answers a question that has been open for about 30 years.

3:00 pm in 345 Altgeld Hall,Thursday, February 17, 2005

Spectral properties of a polyharmonic operator with limit-periodic potential in dimension two

Young-Ran Lee (UIUC Math)

Abstract: We consider a polyharmonic operator: $$ H=(-\Delta)^l+\sum_{n=1}^{\infty} V_n(x),$$ where $V_n(x)$ is periodic with the periods growing exponentially as $2^n$ and the $L_{infty}$-norm decaying super-exponentially. We have shown that when $l>6$, a generalized version of the Bethe-Sommerfeld conjecture holds for this operator, in other words, its spectrum contains a semi-axis. We have proved also that there are eigenfunctions which are close to plane waves. (joint work with Yulia Karpeshina)

4:00 pm in 245 Altgeld Hall,Thursday, February 17, 2005

On Cachazo-Douglas Seiberg-Witten Conjecture for Simple Lie Algebras

Shrawan Kumar (University of North Carolina)