Aspects of aperiodic order

Maria Isabel Cortez: Toeplitz subhifts, equicontinuous systems and residually finite groups
In 1969, Jacobs and Keane introduced Toeplitz subshifts in the context of $\mathbb{Z}$-actions. Since then, different works have shown that there are dynamical systems within this family of subshifts with varied behavior. For example, Downarowicz has shown in 1991 that any possible set of invariant probability measures is realizable by some Toeplitz subshift in $\{0,1\}^{\mathbb{Z}}$. The flexibility of Toeplitz subshifts motivated further generalizations beyond $\mathbb{Z}$-actions, providing examples of group actions on the Cantor set with exciting properties. Another consequence of the generalization of the Toeplitz subshift concept is the characterization of infinite residually finite groups as those admitting actions corresponding to non-periodic Toeplitz subshifts.
In this talk, we gather some results about Toeplitz subshifts and their relationship with minimal equicontinuous systems. For example, applying results concerning Furstenberg-Weiss type almost 1-1 extensions, we will show that Toeplitz subshifts are a test family for the amenability of residually finite groups.

Johannes Kellendonk: Semigroup methods in aperiodic order theory
One of the basic tools to study aperiodic tilings is the theory of ergodic dynamical systems. Properties of tiling dynamical systems reflect properties of the tiling, a point of view which has been very fruitful in the diffraction theory of aperiodic media and also in the description of their topological phases.
We consider a tiling to be highly ordered if its associated dynamical system is “nearly equicontinuous”. By this we mean that the tiling dynamical system factors onto a group rotation so that the fibres of the factor map are finite.
The perhaps simplest class of such tilings is defined by constant shape substitutions on a finite alphabet $A$. In this case the substitution is given by a shape with a collection of maps from $A$ to $A$. These maps define a semigroup whose algebraic structure can be employed to study factors of the dynamical system of various types. This, in turn, is also useful to study another semigroup associated to the dynamical system, namely the enveloping (or Ellis) semigroup.

David Damanik: Deterministic Delocalization
We present joint work with Artur Avila on delocalizing Schrödinger operators in arbitrary dimensions via arbitrarily small perturbations of the potential.