Lecture Series N.34: New Developments on the Discrepancy Function and Related Results (2009)
Principle Speaker: Michael Lacey, Georgia Institute of Technology
Public Lecture by Michael Lacey "The Mathematics of Futurama"
National Science Foundation support for the Spring Lecture Series DMS0751330, March 15, 2008-February 28, 2010, $72345. Luca Capogna lcapogna@comp.uark.edu (Principal Investigator) and Loredana Lanzani (Co-Principal Investigator)
Invited Speakers:
Oleksandra Beznosova (U. Missouri)
Linear with respect to the $A_2$-constant of the weight $w$ bound on the norm of the perfect dyadic operator on the weighted Lebesgue spaces $L^2(w)$.
Abstract: We prove a sharp version of the T(1) theorem for the the perfect dyadic singular integral operator on the real line.
Dmitriy Bilyk (U South Carolina)
Upper bounds in discrepancy theory
Abstract:I will talk about upper bounds in irregularities of distribution, i.e. point sets which have low discrepancy in various senses. I will discuss some classical examples as well as recent developments.
Philip Gressman (U. Pennsylvania)
Generalized moment-entropy equations on the real line.
Abstract: A well-known result in information theory states that, among probability density functions on the real line with a given (finite) variance, the entropy is maximized by a Gaussian. We will consider the related problem of estimating non-oscillatory integrals on the real line from below in terms of quantities related to entropy. The relatively elementary estimates obtained in this way have wide-ranging applications in harmonic analysis, including an improvement of the work of Tao and Wright on 1-dimensional averages.
Mariah Hamel
On Sums of Sets of Primes with Positive Relative Density
Abstract: We will show that if $A$ is a subset of the primes with positive relative density, then $A+A$ must have comparable positive density in $\mathbb{Z}$. Our proof combines Fourier analytic techniques of Green and Green-Tao with a combinatorial result on sumsets of subsets of the multiplicative group of integers modulo $m$. This is joint work with Karsten Chipeniuk.
Nets Katz (U. Indiana)
Discrete analogs of the Kakeya problem.
Abstract: (joint with L. Guth) Using the algebraic method of Dvir, we solve the Joints problem, proving that any set of $N$ lines in $R^3$ describe at most $O(N^{{3 \over 2}})$ joints, where a joint is a point at which three of the lines intersect in a noncoplanar fashion. We solve a related problem of Bourgain.
Alex Iosevich (U. Missouri)
Finite point configurations in vector spaces over finite fields
Abstract:We shall see that if a subset of $F_q^d$ is sufficiently large, then it contains a positive proportion of all finite point configurations up to congruence.
Izabella Laba (U. British Columbia)
The Favard length of product Cantor sets
Abstract: Let K_n be the n-th iteration of a self-similar Cantor set K in the plane. The Favard length Fav(K_n) of K_n is defined as the average length of a 1-dimensional projection of K_n. It is well known that if K has Hausdorff dimension 1, but is not contained in a line, then Fav(K_n) goes to zero as n goes to infinity. The hard problem is to estimate the exact rate of decay. Last year, Nazarov, Peres and Volberg obtained a power-type upper bound for the 4-corner Cantor set. In a joint work with Kelan Zhai, we extend their result to a more general class of product Cantor sets.
Entao Liu
Greedy is Good: Sparse Signal Recovery
Abstract: In the last few years, a new method called compressive sensing has changed the conventional knowledge in acquisition and reconstruction of compressible signals. In this talk, a new idea to weaken a greedy algorithm in compressive sensing will be presented.
Neil Lyall
Polynomial Configurations in Difference Sets
Abstract: I plan to discuss some joint work with Akos Magyar concerning certain arithmetic properties of dense sets of integer points.
Akos Magyar (U. British Columbia)
Singular Radon transforms on discrete nilpotent groups: a model case
Abstract: We study discrete analogues of singular averages associated to polynomial curves on nilpotent groups, focusing on the special case of a cubic curve on the Heisenberg group. This is joint work with A. Ionescu, E.M. Stein and S. Wainger.
Kabe Moen
Sharp Weighted Bounds for Fractional Integral Operators
Abstract. The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant associated to the weights. We obtain analogous results for the fractional integral operator to Petermichl's sharp weighted bounds for singular integral operators. We also obtain analogous results for the fractional integral operator to some difficult open problems in weighted theory for Calderon-Zygmund operators. Our results rely on a sharp o-diagonal version of the extrapolation theorem of Rubio de Francia. As an application of our techniques we obtain improved weighted Sobolev inequalities.
Kostya Oskolkov (U. South Carolina)
On "non-differentiable" function of B. Riemann, and Schroedinger equation
Abstract: In the talk, applications will be presented, of the Schroedinger equation to the study of differential, oscillatory and multi-fractal properties of the functions $$\mathcal R_\alpha=\sum_{n\in \mathbb Z\setminus \{0\}} \frac{e^{\pi i (tn^2+2xn)}{|n|^\alpha}, (t,x)\in \mathbb R^2$$. The imaginary part of $\mathcal R_2$ for x = 0 is $R = \sum_{n\in \mathbb N} n^{-2} \sin \pi n^2 t}$. $R$ was proposed by B. Riemann as a plausible example of a function that is continuous and nowhere differentiable.
Katharine Ott
Spectral properties of the single layer potential operator for the Lame' system of elastostatics on domains with isolated singularities
Abstract: We establish sharp well-posedness results for the regularity problem for the Lam\'{e} system of elastostatics in the class of curvilinear polygons in two dimensions. The key technical ingredient is obtaining spectral properties for the boundary version of the single layer potential operator associated with the Lam\'{e} system acting on $p$-integrable functions on the boundary. Our approach relies on Mellin transform and global optimization techniques. This is joint work with Irina Mitrea and Warwick Tucker.
Malabika Pramanik (U. British Columbia)
Maximal averages over sparse sets in $\mathbb R$
Abstract: (joint with Izabella Laba) We discuss $L^p$ mapping properties of a maximal operator associated with averages against certain singular measures obtained by Cantor type constructions. In particular, this answers a question of Aversa and Preiss concerning density theorems, and extends a result of Rubio de Francia on a generalization of Stein's theorem for the spherical maximal function. Comparisons and contrasts with Bourgain's circular maximal theorem will be explored.
Alexander Stokolos
A gentle introduction in the Bellman Function technique
Abstract: I will explain how the Bellman Function methodology is working using several examples. In particular I will show how to find the Bellman Function for the dyadic maximal operator (joint result with Leonid Slavin and Vasily Vasyunin).
Betsy Stovall
Strong-type endpoint $L^p \to L^q$ estimates for certain generalized Radon transforms
Abstract: A new technique of M. Christ allows one to prove strong-type $L^p \to L^q$ estimates using combinatorial methods which had previously only yielded restricted weak-type bounds. We will discuss this technique and some applications of it.
V. N. Temlyakov (U. South Carolina)
Cubature Formulas, Discrepancy, and Nonlinear Approximation
Abstract: The main goal of this talk is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. In particular, we will show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. We also present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class.
Mingrui Yang
Greedy approximations with regard to multivariate bases with the tensor product structure
Abstract: We compare the greedy algorithm with best $m$-term approximation on classes with regard to various bases. More precisely, we determine the exact error bound for both the greedy approximation and best $m$-term approximation on the class of functions whose coefficients belong to some Lorentz-type sequence space, where the bases can be taken as greedy bases, quasi-greedy bases or the tensor product of greedy bases.