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.

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. 

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.

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.

We establish a quantitative result on norm convergence of triple ergodic averages with respect to three general commuting transformations by proving an r-variation estimate, r > 4, in the norm. We approach the problem via real harmonic analysis, using the recently developed techniques for bounding singular Brascamp-Lieb forms. It is not known whether such norm-variation estimates hold for all r>=2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r<infinity for more than three commuting transformations. This is joint work with Christoph Thiele and Lenka Slavikova.