We prove that negative alternating chainmail links are L-space links. Many of these can be shown to have asymmetric link complements. We also show that augmented chainmail links admit many L-space surgeries, showing that any 3-manifold is surgery on a generalized L-space link. This answers a question of Yajing Liu. We can show that some of these L-spaces have non-orderable fundamental group.

In various cases, a law (that is, a quantifiers free formula) that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability larger than 5/8, must be abelian. For infinite groups, one needs to work a bit harder to define the probability that a given law holds. One natural way is by sampling a random element uniformly from the r-ball in the Cayley graph and taking r to infinity; another way is by sampling elements using random walks. It was asked by Amir, Blachar, Gerasimova, and Kozma whether a law that holds with probability 1, must actually hold globally, for all elements. In a recent joint work with Be'eri Greenfeld, we give a negative answer to their question, using small cancellation methods.

In the talk I will give an overview of probabilistic laws on finitely generated groups, and illustrate how small cancellation methods can reveal unexpected phenomena. For example, I will present a group in which the law x^p=1 holds with probability 1, but yet the group does not satisfy this law, or any other law, globally.

Originating from harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 80s. A set S of natural numbers is an interpolation set for a class of topological dynamical systems C if any bounded sequence on S can be extended to a sequence that arises from a system in C. In this talk, we will provide combinatorial characterizations of interpolation sets for several classes of dynamical systems: minimal systems, weak and strong mixing systems, uniquely ergodic and strictly ergodic systems. We will also talk about an application of interpolation sets to the Sarnak conjecture in multiplicative number theory.

The Torelli group of a closed, oriented surface is the kernel of the action of the mapping class group on the first homology of the surface. Birman asked if the Torelli group is finitely presented for all surfaces of sufficiently large genus. We will a discuss our recent theorem, which says that the second rational homology of the Torelli group is finite dimensional for all closed, oriented surfaces of sufficiently large genus. This result rules out the simplest obstruction to the finite presentability of the Torelli group.

In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.

The Lorentz gas was originally introduced as a model for the movement of electrons in metals. It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers S (atoms of the metal) with elastic collisions at the boundaries ∂S. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood. In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Treviño. We establish some dynamical properties which are common for the periodic and quasiperiodic billiard. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane.