UIUC Dept. of Mathematics Seminar Calendar

     August 2012           September 2012          October 2012    
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Thursday, September 13, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, September 13, 2012

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Submitted by ford.

Dermot McCarthy (Texas A&M)
Hypergeometric Functions over Finite Fields and Modular Forms
Abstract: Hypergeometric functions over finite fields were introduced by Greene in the 1980's as analogues of the classical hypergeometric function. His motivation was to `bring some order' to the area of character sums and their evaluations by appealing to the rich theory of the classical function, and, in particular, its transformation properties. Since then, these finite field hypergeometric functions have also exhibited interesting properties in other areas. In particular, special values of these functions have been related to the Fourier coefficients of certain elliptic modular forms. Relationships with Siegel modular forms of higher degree are also expected. We will outline recent work on proving an example of such a connection, whereby a special value of the hypergeometric function is related to an eigenvalue associated to a Siegel eigenform of degree 2. This is joint work with Matt Papanikolas.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, September 13, 2012

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Submitted by aimo.

Vasileios Chousionis (UIUC Math)
Singular integrals, self-similar sets and removability in the Heisenberg group. (Joint work with P. Mattila)
Abstract: We study singular integrals on lower dimensional subsets of metric groups where the main examples we have in mind are Euclidean spaces and Heisenberg groups. We prove a general unboundedness criterion for singular integrals which extends results in Euclidean spaces to more general kernels than previously considered. Moreover it can be used in order to determine the critical dimension for removable sets of Lipschitz harmonic functions in the Heisenberg group, in an analogous way as in the Euclidean case.

Commutative Ring Theory
3:00 pm in 243 Altgeld Hall, Thursday, September 13, 2012

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Submitted by s-dutta.

Javid Validashti (UIUC Math)
On Syzygies and Singularities of Tensor Product Surfaces
Abstract: On Syzygies and Singularities of Tensor Product Surfaces. Let $U \subseteq H^0({\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}}(2,1))$ be a basepoint free four-dimensional vector space. We study the associated bigraded ideal $I_U \subseteq \textsf{k}[s,t;u,v]$ from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for the image of the projective surface in $\mathbb{P}^3$ parametrized by generators of $U$ over $\mathbb{P}^1 \times \mathbb{P}^1$. This problem arises from a real world application in geometric modeling, where one would like to understand the implicit equation and singular locus of a parametric surface. This talk is based on a joint work with H. Schenck and A. Seceleanu.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, September 13, 2012

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Submitted by kapovich.

David Borthwick (Emory University)
Resonances of hyperbolic surfaces
Abstract: The spectral theory of compact hyperbolic surfaces is an old topic with many interesting results, many of which originate in Atle Selberg's approach to the study of automorphic forms. Selberg's techniques also extend to non-compact surfaces of finite area, although the analysis is somewhat more difficult in this case. For hyperbolic surfaces of infinite area, however, much of the method that was so successful in the compact setting appears to fail. Although the basic spectral properties of such manifolds were worked out in the 1980's by Lax and Phillips, there were no clear infinite-area analogs for the beautiful results of the Selberg theory at that point. This situation started to change in the mid-1990's. Breakthroughs in geometric scattering theory, and in the theory of resonances in particular, have given us in the last 15 years a much more complete picture of the spectral theory of hyperbolic surfaces of infinite area. Many results of the Selberg theory from the compact case do turn out to have very close analogs in this setting, even though the spectral theory is radically different. In this talk we will attempt to give an accessible introduction the spectral theory of hyperbolic surfaces. After highlighting some of the classical results of the Selberg theory, our main goal will be to explain recent developments in the infinite-area case.