Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever we want, to illuminate its entire surface. What is the minimum number of light sources that we need? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). Moreover, there are some rare examples for which a basic, folklore argument could quickly lead to the upper bound 2^n, while at the same time understanding the equality cases has remained elusive for decades. One such example would be convex bodies very close to the cube, which was settled by Livshyts and Tikhomirov in 2017.

In this talk I will discuss another such instance, which comes from the class of 1-unconditional convex bodies, and which also 'forces' us to settle the conjecture for a few more cases of 1-unconditional bodies. This is based on joint work with Wen Rui Sun, and our arguments are primarily combinatorial.

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.