A deep result of Furstenberg from 1967 states that if Γ is a lattice in a semisimple Lie group G, then there exists a measure on Γ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of G is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.
Let SO3(ℝ) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure μ. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO3(ℝ) with sufficiently small measure, then μ(A2) > 3.99 μ(A).
The restriction conjecture, one of the most central problems in harmonic analysis, studies the Fourier transform of functions defined on curved surfaces; specifically, it claims that the level sets of such Fourier transforms are relatively small. The Mizohata-Takeuchi conjecture further studies the shape of these level sets, and in particular the extent to which they can avoid clustering on lines. In this talk we will present a small improvement on the Mizohata-Takeuchi conjecture. This is joint work with Anthony Carbery and Hong Wang.
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=ZM of a finite cyclic group. Coven and Meyerowitz (1998) proposed a characterization of all finite tiles in terms of the cyclotomic divisors of associated mask polynomials, and proved it when the tiling period M has at most two distinct prime factors. In joint work with Itay Londner, we extended it to the case when M=(pqr)2, where p,q,r are distinct primes. The methods we developed can be applied to other questions that hinge on cyclotomic divisibility, ranging from number theory to harmonic analysis and geometric measure theory. In particular, Caleb Marshall and I were able to use cyclotomic divisibility methods to prove new Favard length estimates for product Cantor sets. The talk will provide an introduction to this group of problems.
Morrey’s conjecture arose from a rather innocent-looking question in 1952: is there a local condition characterizing 'ellipticity' in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first progress was made with V. Sverak’s ingenious counterexample. Nevertheless, the case pertaining to planar maps remains open despite much progress, and has fascinated many through its interesting connections to complex analysis, geometric function theory, harmonic analysis, probability and martingales, differential inclusions and the geometry of matrix space. In the talk, I will give an overview of some of these connections and some of the recent progress.
In the absence of measures fully invariant with respect to a group action, this role can be to a certain extent played by the measures "invariant on average", with respect to a certain fixed distribution on the group. These measures are called stationary, and they naturally arise as harmonic measures of random walks. I will provide several partial answers to the general question about the dependence of harmonic measures on the underlying step distributions on the group and discuss counterexamples related to the Minkowski and Denjoy measure classes on the boundary of the classical modular group.
We show that the set of integers n for which n | (2n)!/(n!)2 has a positive asymptotic density, answering a question of Pomerance. The proof uses a mix of ideas from harmonic analysis and the anatomy of integers, and is joint work with Sergei Konyagin.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
An LMS online lecture course in harmonic analysis.
The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations.
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