Carleson proved in 1966 that the Fourier series of any square integrable
function converges pointwise almost everywhere to the function, by establishing boundedness
of the maximally modulated Hilbert transform from L^2 into weak L^2. This
talk is about a generalization of his result, where the Hilbert transform
is replaced by a singular integral operator along a paraboloid. I will
review the history of extensions of Carleson's theorem, and then discuss
the two main ingredients needed to deduce our result: sparse bounds for
singular integrals along the paraboloid, and a square function argument
relying on the geometry of the paraboloid.
A cellular automaton describes a process in which cells evolve
according to a set of rules. Which rule is applied to a specific cell
only depends on the states of the neighboring and the cell itself.
Considering a one-dimensional cellular automaton with finite range,
the positive rates conjecture states that under the presence of
noise the associated stationary measure must be unique. We restrict
ourselves to the case of nearest-neighbor interaction where
simulations suggest that the positive rates conjecture is true. After
discussing a simple criterion to deduce decay of correlations, we show
that the positive rates conjecture is true for almost all
nearest-neighbor cellular automatons. The main tool is comparing a
one-dimensional cellular automaton to a properly chosen
two-dimensional Ising-model. We outline a pathway to resolve the
remaining open cases and formulate a conjecture for general Ising
models with odd interaction.
This presentation is based on collaborative work with Maciej
Gluchowski from the University of Warsaw and Jacob Manaker from UCLA
Sharp geometric and functional inequalities play an important role in analysis, PDEs and differential geometry. In this talk, we will describe our works in recent years on sharp higher order Poincare-Sobolev and Hardy-Sobolev-Maz'ya inequalities on real and complex hyperbolic spaces and noncompact symmetric spaces of rank one. The approach we have developed crucially relies on the Helgason-Fourier analysis on hyperbolic spaces and establishing such inequalities for the GJMS operators. Best constants for such inequalities will be compared with the classical higher order Sobolev inequalities in Euclidean spaces. The borderline case of such inequalities, such as the Moser-Trudinger and Adams inequalities will be also considered.
In mathematical physics, non self-adjoint operators often arise in connection with processes that do not conserve energy. From the mathematical point of view, such operators are of interest because they arise as the quantizations of complex-valued symbols, and the associated classical dynamics must be extended into the complex domain. In this talk, I will discuss the special class of non self-adjoint pseudodifferential operators with double characteristics, and I will present some new results on L^p-bounds for eigenfunctions of such operators in the semiclassical limit. The main tools used are the Fourier-Bros-Iagolnitzer (FBI) transform and microlocal analysis in exponentially weighted spaces of holomorphic functions.
The Brascamp-Lieb inequality and the reverse form generalize the Holder and Prekopa-Leindler inequality. The equality case in the Brascamp-Lieb inequality has been characterized by Valdimarsson. Partially building on the work of Bennett, Carbery, Christ and Tao we characterize the equality case in the Reverse Brascamp-Lieb inequality. The proof builds on the structure theory of ‘’Brascamp-Lieb data’’ and uses a variant of Caffarelli's contraction principle. We will also discuss some geometric applications, concerning volume estimates from orthogonal projections and sections. This is based on joint work with Karoly Boroczky and Dongmeng Xi.
Remez-type inequalities bound the suprema of low-degree polynomials over some domain K by their suprema over a subset S of K. Existing multi-dimensional Remez inequalities bear constants with strong dependence on dimension. In this talk we will prove a dimension-free Remez-type estimate when K is the polydisc D^n and S is from a certain class of discrete subsets. As a direct consequence we also obtain a Bohnenblust-Hille-type inequality for products of cyclic groups, which in turn has consequences for learning algorithms. Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.
This talk is about maximal averages on spheres in two and
higher-dimensional Euclidean space. This is a classic topic in harmonic analysis originating in questions
on differentiability properties of functions. We consider maximal spherical averages with a supremum taken over a
given dilation set. It turns out that the sharp Lp improving properties of such operators are closely related
to fractal dimensions of the dilation set such as the Minkowski and Assouad dimensions.
This leads to a surprising characterization of the closed convex sets which can occur as closure of the sharp Lp improving region of such a maximal operator. This is joint work with Andreas Seeger. If time allows we will also mention recent work on the Heisenberg group and related work in progress.
Consider the n-cube graph in R^n, with vertices {0,1}^n and edges connecting vertices with Hamming distance 1.
How many hyperplanes are required in order to dissect all edges?
This problem has been open since the 70s. We will discuss this and related problems.
Puzzle: Show that n hyperplanes are sufficient, while sqrt(n) are not enough.