Abstract: In this talk we will deal with multiple ergodic averages having iterates a common integer-valued sequence that comes from appropriate classes of functions. In particular, we are dealing with the Hardy field case. The main part of this talk relies on a joint work with Sebastián Donoso (Universidad de Chile) and Wenbo Sun (Virginia Tech).

Abstract: In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi involution on sets is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. In this setting, certain transport schemes are prohibited and one gets a necessary compatibility condition on measures, which in a special discrete case can be interpreted as Hall’s marriage theorem. 

We will discuss how this unified point of view on order reversing quasi involutions helps to deeper the understanding of the underlying structures and principles, and present many examples, in particular, from convexity. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

Abstract: Many problems first posed in the integers lattice or in Euclidean space have been studied in vector spaces over finite fields. Part of the reason is that finite fields are thought to provide a model setting that is technically easier to analyse. The talk will survey this topic with a slight lean towards examples where the finite field model has limitations (and may be of interest to analysts).

Abstract: In my talk I will introduce a recent construction of tilings of the hyperbolic upper half-space H^{d+1}. These tilings may be viewed as extensions of hyperbolic tilings considered by Boroczky, Penrose and Kamae, and essentially illustrated by Escher, as well as hyperbolic liftings of the d-dimensional Euclidean multiscale substitution tilings introduced in our earlier work. I will describe our results about the geodesic and horospheric actions on the associated space of hyperbolic tilings and discuss a prime orbit theorem for the geodesic flow. Based on joint work in progress with Yaar Solomon.  

Abstract: Substitution systems are one important source for mathematical quasicrystals, i.e. well-scattered point sets in a space that are non-periodic, but still exhibit some long range order. We present a geometric framework in order to construct symbolic configurations ("colorings") over lattices in homogeneous Lie groups, arising as fixed points of some substitution rule. Assuming mild conditions, one obtains non-periodic configurations with strictly ergodic hull system. We explain how this leads to first explicit aperiodic examples for linearly repetitive Delone sets in certain dilation groups, such as the Heisenberg group. Joint work with Siegfried Beckus and Tobias Hartnick .

Abstract:  In this talk we will consider maximal operators on R^n formed by taking arbitrary rotations of tensor products of an (n − 1)-dimensional H"ormander-Mihlin multiplier with the identity in one coordinate. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sj"olin’s generalization of Carleson’s maximal operator.

The main result is a weak-type L^2(R^n)-estimate on band-limited functions. As corollaries, we obtain a sharp L^2(R^n) estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set and a version of the Carleson-Sj"olin theorem. Our approach relies on higher dimensional time-frequency analysis elaborated according to the directional nature of the operator under study. This is joint work with Odysseas Bakas, Francesco Di Plinio, and Ioannis Parissis.

Abstract:  In this talk, we prove the natural analogue of the Kakeya conjecture in the alternative ambient space Q_p^n, and we discuss some philosophical similarities and differences between this variant of the Kakeya conjecture and the classical Kakeya conjecture in R^n.

Abstract: In this talk, I will discuss some recent work on area and Hausdorff dimension of projections and intersections in the Heisenberg group. In Euclidean space, it is known that projections of sets onto k-dimensional subspaces almost surely do not decrease Hausdorff dimension, and that projections of sets of dimension greater than k have projections almost surely of positive k-dimensional area. It has been conjectured that these theorems extend to "vertical projections" in the Heisenberg group. This conjecture is still open, but was recently solved in a significant part of the range by Fassler and Orponen, using a "point-plate incidence" method. I will outline some of my recent work, which also uses the point-plate incidence method, and which proves the "positive area" part of the conjecture. One connection of this talk to harmonic analysis is that it uses the (endpoint) trilinear Kakeya inequality, which grew out of multilinear Fourier analysis inspired by the Fourier restriction and Kakeya conjectures. 

Abstract: A Kakeya set is a set of points in R^n which contains a unit line segment in every direction. The Kakeya conjecture states that the dimension of any Kakeya set is n. This conjecture remains wide open for all n \geq 3. Together with Josh Zahl, we study a special collection of the Kakeya sets, namely the sticky Kakeya sets, where the line segments in nearby directions stay close. We prove that sticky Kakeya sets in R^3 have dimension 3,  based on ideas of Katz and Tao and some recent work on projection theorems in geometric measure theory.

Abstract: Kamke solved an analog of Waring's problem with $n$th powers replaced by integer-valued polynomials. Larsen and Nguyen explored the view of algebraic groups as a natural setting for Waring's problem. In this talk, we will develop a theory of polynomial maps from nonempty commutative semigroups to arbitrary groups, prove that it has desirable formal properties when the target group is locally nilpotent, and apply it to solve an analog of Waring's problem for the general discrete Heisenberg groups $H_{2n+1}(\mathbb{Z})$ for any integer $n\ge 1$. 

Abstract: One of the simplest additive structures is a three-term arithmetic progression. As such, it is a very natural question to ask what kind of conditions on a set are sufficient to force the existence of a three-term progression. This story began in 1953 with Roth's celebrated result that any set of integers of positive density contains a three-term progression. In the 70 years since there has been a great deal of progress in understanding exactly what the correct density threshold for this problem is. In February there was a spectacular breakthrough due to Kelley and Meka, who lowered the upper bound down significantly, almost to the known lower bound. As such a full understanding of three-term progressions at last seems within reach. In this survey talk I will discuss some of the historical developments and the ideas behind the breakthrough Kelley-Meka result.