UIUC Dept. of Mathematics Seminar Calendar

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Thursday, October 13, 2011

BCDE Math/Physics Seminar
12:00 pm in 464 Loomis Laboratory, Thursday, October 13, 2011

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Submitted by katz.

Sue Tolman (UIUC Math)
Equivariant cohomology and the Duistermaat-Heckman function

Harmonic Analysis and Differential Equations
1:00 pm in 445 Altgeld Hall, Thursday, October 13, 2011

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Submitted by laugesen.

Daniel Grieser [email] (U. Oldenburg)
The plasmonic eigenvalue problem
Abstract: (Note the UNUSUAL DAY AND ROOM.) Plasmonics is a modern area of physics where one studies the interaction of electromagnetic waves with the free electrons of a metal ("plasma"). One of the fundamental problems is to determine the resonance frequencies at which this interaction is large. Mathematically this leads to a boundary value problem for harmonic functions in which the resonance frequency is a parameter in the boundary conditions. In the talk I will explain the problem and some results about the resonance frequencies.

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, October 13, 2011

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Submitted by kapovich.

Brian Ray (UIUC Math)
Spectral Rigidity in Free Groups.
Abstract: We say a subset $\Sigma \subseteq F_N$ of the free group of rank $N$ is \emph{spectrally rigid} if whenever $T_1, T_2 \in cv_N$ are $\mathbb{R}$-trees in (unprojectivized) outer space for which $\| \sigma \|_{T_1} = \| \sigma \|_{T_2}$ for every $\sigma \in \Sigma$, then $T_1 = T_2$ in $cv_N$. Recent results by Carette, Francaviglia, Kapovich, and Martino motivate the following question: is it true that for any $H \leq Aut(F_N)$ either for every $1 \neq g \in F_N$, the orbit $Hg$ is rigid or for every $1 \neq g \in F_N$, the orbit $Hg$ is not rigid? We will explain why, in the case $H$ is cyclic, the orbit $Hg$ is never rigid. We will also show that if we impose additional structure on the set $\Sigma$, then we can establish an equivalence between spectral rigidity and a certain ``weak aperiodicity'' property.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, October 13, 2011

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Pekka Pankka (University of Helsinki, Finland)
Quasisymmetric (non-)parametrization of Semmes spaces
Abstract: In ``Thirty-three yes or no questions about mappings, measures, and metrics'' Heinonen and Semmes asked whether a decomposition space $\mathbb R^3/\mathrm{Bd}$ associated to Bing's double admits an Ahlfors $3$-regular and locally linearly contractible metric so that the product space $\mathbb R^3/\mathrm{Bd} \times \mathbb R^m$ is not quasisymmetrically equivalent to $\mathbb R^{3+m}$. This question has its origin in the program of understanding the quasiconformal geometry and analysis on metric spaces homeomorphic to Euclidean spaces and, in particular, in a result of Semmes showing that such good metrics without good parametrizations exist for $\mathbb R^3/\mathrm{Bd}$. Recently Heinonen and Wu answered positively to a related question of Heinonen and Semmes on the stabilized decomposition spaces $\mathbb R^3/\mathrm{Wh} \times \mathbb R^m$ associated with the Whitehead continuum. In this talk, I will discuss Semmes type geometrization of more general classes of decomposition spaces $\mathbb R^3/G$ and quasisymmetric (non-)parametrizability results for $\mathbb R^3/G\times \mathbb R^m$, that include the topologically self-similar cases of Whitehead continuum and Bing's double. This is a joint work with Jang-Mei Wu.

Graduate Analysis Seminar
3:00 pm in 345 Altgeld Hall , Thursday, October 13, 2011

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Vyron Vellis (UIUC Math)
On Rohde snowflakes
Abstract: In his celebrated paper "Quasiconformal Reflections", Ahlfors showed, among other results, that every planar quasicircle is the image of the unit circle under a global quasiconformal map which is also locally uniformly bi-Lipschitz (after scaling) off the quasicircle. We will give a constructive proof of this theorem in the case of some specific fractal sets known as Rohde snowflakes; we then use a result of S. Rohde to extend the proof to all quasicircles. If time permits I will also discuss almost sure dimension estimates of some random Rhode snowflakes. This talk serves as an introduction to some methods used in Quasiconformal Analysis and no specific background is required.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, October 13, 2011

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Submitted by kapovich.

Trevor Wooley (University of Bristol)
Vinogradov�s mean value theorem: old and new estimates with new and old applications
Abstract: Exponential sums of large degree play a prominent role in the analysis of problems spanning the analytic theory of numbers. In 1935, I. M. Vinogradov devised a method for estimating their mean values very much more efficient than the methods available hitherto due to Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free region of the Riemann zeta function, in Diophantine approximation, and in Waring�s problem. Recent applications from the 21st century include sum-product estimates in additive combinatorics, and the investigation of the geometry of moduli spaces. Over the past 75 years, estimates for the moments underlying Vinogradov�s mean value theorem have failed to achieve those conjectured by a factor of roughly log k in the number of implicit variables required to successfully analyse exponential sums of degree k. Recent work comes within a stone�s throw of the best possible conclusions. Aside from indicating the ideas underlying our recent progress on Vinogradov's mean value theorem, we will discuss several applications, some crusty and old, others shiny and new, both in number theory and beyond.