UIUC Dept. of Mathematics Seminar Calendar

      June 2012              July 2012             August 2012     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2    1  2  3  4  5  6  7             1  2  3  4
  3  4  5  6  7  8  9    8  9 10 11 12 13 14    5  6  7  8  9 10 11
 10 11 12 13 14 15 16   15 16 17 18 19 20 21   12 13 14 15 16 17 18
 17 18 19 20 21 22 23   22 23 24 25 26 27 28   19 20 21 22 23 24 25
 24 25 26 27 28 29 30   29 30 31               26 27 28 29 30 31

Thursday, January 19, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, January 19, 2012

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Atul Dixit (UIUC Math)
Convexity of quotients of theta functions
Abstract: For fixed $u$ and $v$ such that $0 \leq u < v < 1/2 $, the monotonicity of the quotients of Jacobi theta functions, namely, $\theta_{j}(u|i\pi t)/\theta_{j}(v|i\pi t)$, $j=1, 2, 3, 4$, on $0 < t < \infty$ has been established in the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the author. In the present paper, we show that the quotients $\theta_{2}(u|i\pi t)/\theta_{2}(v|i\pi t)$ and $\theta_{3}(u|i\pi t)/\theta_{3}(v|i\pi t)$ are convex on $0 < t < \infty$. This is joint work with Arindam Roy and Alexandru Zaharescu.

Thursday, January 26, 2012

Thursday, February 2, 2012

Thursday, February 9, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, February 9, 2012

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Kevin Wildrick (University of Bern)
Lipschitz constants and differentiability almost everywhere
Abstract: Rademacher's theorem that Lipschitz functions are differentiable almost everywhere forms the backbone of results in function theory, geometric measure theory, and geometric topology. A prime example is Cheeger's theorem regarding the existence of differentiable structures on metric spaces supporting a Poincaré inequality. We will review some classical results and then discuss a version of Rademacher's theorem for the "lower" Lipschitz constant, which detects oscillation only on some sequence of scales tending to zero, rather than on all sequences of scales tending to zero. We also provide an example showing the sharpness of the results and the relationship of differentiability to the capacity of points.

Thursday, February 16, 2012

Thursday, March 1, 2012

Thursday, March 29, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, March 29, 2012

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Dean Baskin (Northwestern)
Asymptotics of radiation fields in asymptotically Minkowski space
Abstract: Radiation fields are (appropriately rescaled) limits of solutions of wave equations along light rays. In this talk I will describe a class of (non-static) asymptotically Minkowski space times for which the radiation field is defined and indicate how methods of Vasy can be used to express the asymptotics in terms of the resonances of a related Riemannian problem on an asymptotically hyperbolic manifold. In particular, even on Minkowski space, these methods give a new understanding of the Klainerman-Sobolev estimates. This is joint work with Andras Vasy and Jared Wunsch.

Thursday, April 5, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, April 5, 2012

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Pekka Pankka (University of Helsinki, Finland)
Almost sure limits of quasiconformally equivalent closed manifolds
Abstract: Suppose $(M_i,p_i)$ is a sequence of pointed closed Riemannian manifolds so that the diameters of $M_i$ grow without bounds. By Gromov's compactness theorem, under conditions on curvature and injectivity radius, this sequence has a subsequence converging in the pointed Gromov-Hausdorff topology to a pointed Riemannian manifold $(X,p)$. But what is $X$ typically like? Under the additional condition that manifolds $M_i$ are uniformly quasiconformally equivalent to a fixed manifold, $X$ is almost surely (in a suitable sense) quasiconformal to either the Euclidean space or a punctured Euclidean space. In this talk I will discuss this and similar results for surfaces and graphs and the relation of these results to the work of Benjamini and Schramm. This is joint work with Hossein Namazi and Juan Souto.

Thursday, April 12, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, April 12, 2012

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Kai Rajala (University of Jyväskylä, Finland)
Optimal assumptions for discreteness
Abstract: A celebrated theorem of Reshetnyak's says that a non-constant Sobolev mapping F of R^n is discrete and open if K_F=|DF|^n/det(DF) is uniformly bounded. In view of applications to non-linear elasticity theory, geometric function theory, and certain PDE:s, it is desirable to find the minimal analytic assumptions under which one can conclude discreteness and openness. In dimension two, Iwaniec and Sverak proved that it suffices to assume K_F to be locally integrable. We discuss higher-dimensional versions of the Iwaniec-Sverak theorem due to Manfredi and Villamor, and others, and present our joint work with Stanislav Hencl, showing that the expected sharp higher-dimensional version does not hold.

Thursday, April 26, 2012

Graduate Analysis Seminar
5:00 pm in 147 Altgeld Hall, Thursday, April 26, 2012

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Rami Luisto (University of Helsinki)
On the non-existence of BLD-mappings between manifolds
Abstract: I will give my talk about the main result of my Master's thesis [1]. The result can be seen to be a metric version of the Varopoulos theorem, which states that if N is a compact n-dimensional Riemannian manifold whose fundamental group has hyperbolic growth rate, then there exists no quasiregular mapping from the Euclidean n-space to N. In my thesis I translate this result to the metric setting by talking about path-metric manifolds and Bounded Length Distortion (BLD) mappings between them. A BLD mapping, in short, is an open, discrete and continuous mapping that preserves the lengths of rectifiable paths up to a fixed multiplicative constant. In my talk I will introduce the concepts of the growth rate of a finitely generated group, path-length structure of a manifold and the basic properties of Bounded Length Distortion mappings. If we have time left, I will talk about some results of my Licentiate's thesis which embetter the results given in my Master's thesis. There might be chocolate available during the presentation.

[1] Luisto, Rami. ``On the non-existence of BLD-mappings between manifolds.'' Master's thesis, available at http://helsinki.fi/~luisto/ProGradu.pdf

Friday, September 7, 2012

Analysis Seminar
2:00 pm in 241 Altgeld Hall, Friday, September 7, 2012

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Gaven Martin (Massey University, New Zealand)
The solution to Siegel's Problem
Abstract: We outline the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram. This solves (in three dimensions) the problem posed by Siegel in 1945. Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area. There are strong connections with arithmetic hyperbolic geometry in the proof and the result has applications in the maximal symmetry groups of hyperbolic 3-manifolds (in much the same way that Hurwitz 84g-84 theorem and Siegel's result do).

Thursday, September 13, 2012

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

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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.

Thursday, October 18, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, October 18, 2012

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Raanan Schul (SUNY Stony Brook)
Lipschitz Functions vs. Projections
Abstract: We discuss joint work with Jonas Azzam. ``All Lipschitz maps from $R^7$ to $R^3$ are orthogonal projections''. This is of course quite false as stated. It turns out however, that there is a surprising grain of truth in this statement. We show that all Lipschitz maps from $R^7$ to $R^3$ (with 3-dimensional image) can be precomposed with a map $g:R^7\to R^7$ such that $f\circ g$ will satisfy, when we write the domain as $R^4\times R^3$ and restrict to $E$, a large portion of the domain, that $f\circ g$ will be constant in the first coordinate and biLipschitz in the second coordinate. Geometrically speaking, the map $g$ distorts $R^7$ in a controlled manner, so that the fibers of $f$ are straightened out. Our results are quantitative. The target space can be replaced by any metric space! The size of the set $E$ on which behavior is good is an important part of the discussion and examples such as Kaufman's 1979 construction of a singular map $[0,1]^3$ onto $[0,1]^2$ are an important enemy. On route we will discuss a new extension theorem which is used to construct the bilipschitz map $g$, improving results of Jones (88) and David (88). In particular, if $g:R^7\to R^7$ is a Lipschitz map, then it agrees with a globally defined biLipschitz map $\hat{g}:R^7\to R^7$ on a large piece of the domain. This was previously known only by increasing the dimension of the target space of $\hat{g}$ (David and Semmes, 91).

Thursday, October 25, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, October 25, 2012

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Valentino Magnani (University of Pisa, Italy)
On the sub-Riemannian measure of submanifolds
Abstract: We present an explicit formula for the sub-Riemannian measure of a submanifold, embedded in a fixed stratified group. We show that in many cases this measure is equivalent to the Hausdorff measure of the submanifold with respect to the sub-Riemannian distance of the group. We discuss the important class of transversal submanifolds, for which the equivalence between Hausdorff measure and sub-Riemannian measure holds. This result has been recently obtained in collaboration with Jeremy Tyson and Davide Vittone. Some open questions will be addressed.

Thursday, November 8, 2012

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, November 8, 2012

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Alexandra Kolla (UIUC Computer Science)
Maximal Inequality for Spherical Means on the Hypercube
Abstract: In this talk, we establish a dimension-free ell_2 maximal inequality for spherical means on the Hypercube graph {0,1}^n. We present possible connections to a key open problem in theory of computer science, namely, the Unique Games Conjecture. Combinatorially, this inequality implies the following stronger alternative to the union-bound technique: Assume that we have a binary function f (values 0,1) on the n-dimensional hypercube Hn, with N=2^n vertices. Think of the set X={x\in Hn : f(x)=1} as the set vertices that a malicious adversary "spoils". Assume |X|<\epsilon N, i.e. the adversary can only spoil up to \epsilon fraction of the vertices. Fix a threshold \lambda>\epsilon. Say a (hamming) sphere S(x,r) or radius r around a point x is "bad" if the adversary has spoiled more than \lambda fraction of the points in the shpere. We call a point x "ruined" if *there exists* a radius r for which the sphere S(x,r) around x is bad. Our maximal inequality implies that for every \lambda, there is an absolute constant \epsilon (which does not depend on the dimension n) that if the adversary spoils at most \epsilon fraction of the points then the "ruined" set is a strict subset of the hypercube. Note that applying a union-bound over radii instead, we would not get any useful inequality for the size of the ruined set. Joint work with Aram Harrow (UW) and Leonard Schulman (Caltech).

Thursday, November 15, 2012

Thursday, December 6, 2012