Seminars and Colloquia by Series

Commutators and BMO

Series
Analysis Seminar
Time
Wednesday, March 2, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Brett Wick – GT and Washington University St Louis
In this talk we will discuss the connection between functions with bounded mean oscillation (BMO) and commutators of Calderon-Zygmund operators. In particular, we will discuss how to characterize certain BMO spaces related to second order differential operators in terms of Riesz transforms adapted to the operator and how to characterize commutators when acting on weighted Lebesgue spaces.

Multicommutators

Series
Analysis Seminar
Time
Wednesday, February 24, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Danqing He – University of Missouri, Columbia –
We generalize the Calderon commutator to the higher-dimensional multicommutator with more input functions in higher dimensions. For this new multilinear operator, we establish the strong boundedness of it in all possible open points by a new multilinear multiplier theorem utilizing a new type of Sobolev spaces.

Asymptotic zero distribution of some multiple orthogonal polynomials

Series
Analysis Seminar
Time
Monday, February 22, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Walter Van Assche – University of Leuven, Belgium
The asymptotic distribution of the zeros of two families of multiple orthogonal polynomials will be given, namely the Jacobi-Pineiro polynomials (which are an extension of the Jacobi polynomials) and the multiple Laguerre polynomials of the first kind (which are an extension of the Laguerre polynomials). We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution.

The role of VC-dimension in the one-bit restricted isometry property

Series
Analysis Seminar
Time
Wednesday, February 17, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Scott Spencer – Georgia Tech –
Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.

Characterization of two parameter matrix-valued BMO

Series
Analysis Seminar
Time
Wednesday, February 10, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dario Mena – Georgia Tech –
In this work we prove that the space of two parameter, matrix-valued BMO functions can be characterized by considering iterated commutators with the Hilbert transform. Specifically, we prove that the norm in the BMO space is equivalent to the norm of the commutator of the BMO function with the Hilbert transform, as an operator on L^2. The upper bound estimate relies on a representation of the Hilbert transform as an average of dyadic shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's wavelet proof for the scalar case.

A Discrete Quadratic Carleson Theorem

Series
Analysis Seminar
Time
Wednesday, February 3, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Lacey – Gatech
We will describe sufficient conditions on a set $\Lambda \subset [0,2\pi) $ so that the maximal operator below is bound on $\ell^2(Z)$. $$\sup _{\lambda \in \Lambda} \Big| \sum_{n\neq 0} e^{i \lambda n^2} f(x-n)/n\Big|$$ The integral version of this result is an influential result to E.M. Stein. Of course one should be able to take $\Lambda = [0,2\pi) $, but such a proof would have to go far beyond the already complicated one we will describe. Joint work with Ben Krause.

One Bit Sensing, RIP bounds and Empirical Processes

Series
Analysis Seminar
Time
Wednesday, January 27, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Lacey – Gatech
A signal is a high dimensional vector x, and a measurement is the inner product . A one-bit measurement is the sign of . These are basic objects, as will be explained in the talk, with the help of some videos of photons. The import of this talk is that one bit measurements can be as effective as the measurements themselves, in that the same number of measurements in linear and one bit cases ensure the RIP property. This is explained by a connection with variants of classical spherical cap discrepancy. Joint work with Dimtriy Bilyk.

Exponential bases and frames on fractals

Series
Analysis Seminar
Time
Wednesday, January 20, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
ChunKit Lai – San Francisco State University –
We study the construction of exponential bases and exponential frames on general $L^2$ space with the measures supported on self-affine fractals. This problem dates back to the conjecture of Fuglede. It lies at the interface between analysis, geometry and number theory and it relates to translational tilings. In this talk, we give an introduction to this topic, and report on some of the recent advances. In particular, the possibility of constructing exponential frames on fractal measures without exponential bases will be discussed.

Orthogonal polynomials for the Minkowski question Mark function

Series
Analysis Seminar
Time
Wednesday, December 2, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Walter Van Assche – University of Leuven, Belgium
The Minkowski question mark function is a singular distribution function arising from Number Theory: it maps all quadratic irrationals to rational numbers and rational numbers to dyadic numbers. It generates a singular measure on [0,1]. We are interested in the behavior of the norms and recurrence coefficients of the orthonormal polynomials for this singular measure. Is the Minkowski measure a "regular" measure (in the sense of Ullman, Totik and Stahl), i.e., is the asymptotic zero distribution the equilibrium measure on [0,1] and do the n-th roots of the norm converge to the capacity (which is 1/4)? Do the recurrence coefficients converge (are the orthogonal polynomials in Nevai's class). We provide some numerical results which give some indication but which are not conclusive.

Mixed norm Leibnitz rules via multilinear operator valued multipliers

Series
Analysis Seminar
Time
Thursday, November 19, 2015 - 16:35 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Francesco Di Plinio – Brown University –
[Special time and location] The content of this talk is joint work with Yumeng Ou. We describe a novel framework for the he analysis of multilinear singular integrals acting on Banach-valued functions.Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces, including, in particular, noncommutative Lp spaces. A concrete case of our result is a multilinear generalization of Weis' operator-valued Hormander-Mihlin linear multiplier theorem.Furthermore, we derive from our main result a wide range of mixed Lp-norm estimates for multi-parameter multilinear multiplier operators, as well as for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform. These respectively extend the results of Muscalu et. al. and of Silva to the mixed norm case and provide new mixed norm fractional Leibnitz rules.

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