We consider ball averages on discrete groups, and associated Hardy-Littlewood maximal operator, with the balls defined by invariant metrics associated with a variety of length functions. Under natural assumptions on the rough radial structure of the group under consideration, we establish a maximal inequality of weak-type for the Hardy-Littlewood operator. These assumptions are related to a coarse radial median inequality, to almost exact polynomial-exponential growth of balls, and to the rough radial rapid decay property. We give a variety of examples where the rough radial structure assumptions hold, including any lattice in a connected semisimple Lie group with finite center, with respect to the Riemannian distance on symmetric space restricted to an orbit of the lattice. Other examples include right-angled Artin groups, Coxeter groups and braid groups, with a suitable choice of word metric. For non-elementary word-hyperbolic groups we establish that the Hardy-Littlewood maximal operator with respect to balls defined by a word metric satisfies the weak-type (1,1)-maximal inequality, which is the optimal result. This is joint work with Koji Fujiwara, Kyoto University.

Hyperplane arrangements cut space into `regions', which we like to count. Although all regions are $n$-dimensional, some are more bounded than others, captured by the `level' of a region. Can we refine our region counting by level? And how do level counts interact with other properties of the arrangement? This talk should be highly approachable, requiring only the ability to visualize high-dimensional objects.

The Ramsey number R(t, k) is the smallest n such that any red-blue edge coloring of the n-vertex complete graph has either a t-vertex red complete subgraph or a kvertex blue complete subgraph. We will investigate the history of asymptotic bounds on the extreme off-diagonal Ramsey number R(3, k), and present a new lower bound that has been conjectured to be asymptotically tight. Based on joint work with Paul Horn, Dylan King, Florian Pfender.

Our set up will consist of a countable group acting on a metric space with dense orbits. Our goal will be to develop effective gauges that measure how dense such orbits actually are, or equivalently how efficient is the approximation of a general point in the space by the points in the orbit.  We will describe several such gauges, whose definitions are motivated by classical Diophantine approximation, and are related to approximation exponents, discrepancy and equidistribution. We will then describe some of the (non-classical) examples we aim to analyze, focusing mainly on certain countable subgroups of the special linear or affine group, or of the groups of isometries of hyperbolic spaces, acting on some associated homogeneous spaces. In this set-up it is possible to establish optimal effective Diophantine approximation results in certain cases. We will very briefly indicate some ingredients of the methods involved, keeping the exposition as accessible as possible. We will also indicate some of the many challenging open problems that this circle of questions present. Based partly on previous joint work with Anish Ghosh and Alex Gorodnik, and partly on recent work with Mikolaj Fraczyk and Alex Gorodnik. 

TBA