Seminars and Colloquia by Series

Maximal averages and singular integrals along vector fields in higher dimension

Series
Analysis Seminar
Time
Friday, September 22, 2017 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Francesco Di PlinioUniversity of Virginia
It is a conjecture of Zygmund that the averages of a square integrable function over line segments oriented along a Lipschitz vector field on the plane converge pointwise almost everywhere. This statement is equivalent to the weak L^2 boundedness of the directional maximal operator along the vector field. A related conjecture, attributed to Stein, is the weak L^2 boundedness of the directional Hilbert transform taken along a Lipschitz vector field. In this talk, we will discuss recent partial progress towards Stein’s conjecture obtained in collaboration with I. Parissis, and separately with S. Guo, C. Thiele and P. Zorin-Kranich. In particular, I will discuss the recently obtained sharp bound for the Hilbert transform along finite order lacunary sets in two dimensions and possible higher dimensional generalization

Sparse Bounds for Discrete Spherical Maximal Averages

Series
Analysis Seminar
Time
Wednesday, September 20, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert KeslerGeorgia Tech
Magyar, Stein, and Wainger proved a discrete variant in Zd of the continuous spherical maximal theorem in Rd for all d ≥ 5. Their argument proceeded via the celebrated “circle method” of Hardy, Littlewood, and Ramanujan and relied on estimates for continuous spherical maximal averages via a general transference principle. In this talk, we introduce a range of sparse bounds for discrete spherical maximal averages and discuss some ideas needed to obtain satisfactory control on the major and minor arcs. No sparse bounds were previously known in this setting.

Some Recent Sparse Bounds

Series
Analysis Seminar
Time
Wednesday, September 13, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LaceyGeorgia Tech
A sparse bound is a novel method to bound a bilinear form. Such a bound gives effortless weighted inequalities, which are also easy to quantify. The range of forms which admit a sparse bound is broad. This short survey of the subject will include the case of spherical averages, which has a remarkably easy proof.

Finite dimension Balian-Low type theorems

Series
Analysis Seminar
Time
Wednesday, September 6, 2017 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shahaf NitzanGeorgia Tech
The classical Balian-Low theorem states that if both a function and it's Fourier transform decay too fast then the Gabor system generated by this function (i.e. the system obtained from this function by taking integer translations and integer modulations) cannot be an orthonormal basis or a Riesz basis.Though it provides for an excellent `thumbs--rule' in time-frequency analysis, the Balian--Low theorem is not adaptable to many applications. This is due to the fact that in realistic situations information about a signal is given by a finite dimensional vector rather then by a function over the real line. In this work we obtain an analog of the Balian--Low theorem in the finite dimensional setting, as well as analogs to some of its extensions. Moreover, we will note that the classical Balian--Low theorem can be derived from these finite dimensional analogs.

Zak transform analysis of shift-invariant subspaces

Series
Analysis Seminar
Time
Wednesday, August 23, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joey IversonUniversity of Maryland
Abstract: Shift-invariant (SI) spaces play a prominent role in the study of wavelets, Gabor systems, and other group frames. Working in the setting of LCA groups, we use a variant of the Zak transform to classify SI spaces, and to simultaneously describe families of vectors whose shifts form frames for the SI spaces they generate.

Self-similar tilings of General Fractal Blow-ups and Anderson Putnam Theory

Series
Analysis Seminar
Time
Wednesday, June 21, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael F. BarnsleyAustralian National University
In this seminar I will discuss current work, joint with AndrewVince and Alex Grant. The goal is to tie together several related areas, namelytiling theory, IFS theory, and NCG, in terms most familiar to fractal geometers.Our focus is on the underlying code space structure. Ideas and a conjecture willbe illustrated using the Golden b tilings of Robert Ammann

Density theorem for continuous frames and the uncertainty principle

Series
Analysis Seminar
Time
Wednesday, April 19, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mishko MitkovskiiClemson University
A well-known elementary linear algebra fact says that any linear independent set of vectors in a finite-dimensional vector space cannot have more elements than any spanning set. One way to obtain an analog of this result in the infinite dimensional setting is by replacing the comparison of cardinalities with a more suitable concept - which is the concept of densities. Basically one needs to compare the cardinalities locally everywhere and then take the appropriate limits. We provide a rigorous way to do this and obtain a universal density theorem that generalizes many classical density results. I will also discuss the connection between this result and the uncertainty principle in harmonic analysis.

Falconer type theorems for simplices

Series
Analysis Seminar
Time
Wednesday, April 12, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eyvi PalssonVirginia Tech
Finding and understanding patterns in data sets is of significant importance in many applications. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will present recent progress on Falconer type problems for simplices. The main techniques used come from analysis and geometric measure theory.

Bounding marginals of product measures

Series
Analysis Seminar
Time
Wednesday, April 5, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Tech
It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This bound yields a sharp small ball inequality for the length of a projection of a random vector. This talk is based on the joint work with G. Paouris and P. Pivovarov.

Persistence as a spectral property

Series
Analysis Seminar
Time
Wednesday, March 29, 2017 - 02:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shahaf NitzanGeorgia Tech
A Gaussian stationary sequence is a random function f: Z --> R, for which any vector (f(x_1), ..., f(x_n)) has a centered multi-normal distribution and whose distribution is invariant to shifts. Persistence is the event of such a random function to remain positive on a long interval [0,N]. Estimating the probability of this event has important implications in engineering , physics, and probability. However, though active efforts to understand persistence were made in the last 50 years, until recently, only specific examples and very general bounds were obtained. In the last few years, a new point of view simplifies the study of persistence, namely - relating it to the spectral measure of the process. In this talk we will use this point of view to study the persistence in cases where the spectral measure is 'small' or 'big' near zero. This talk is based on Joint work with Naomi Feldheim and Ohad Feldheim.

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