Lectures

Titles of the mini-courses are purely indicative, and may be subject to modification.

 

Laurent Bartholdi (ENS, Paris and University of Göttingen):

Self-similar groups.

Abstract:
Self-similar groups are groups defined by a "self-similar action": an action on a set, such that actions on small parts of the set mimick the action on the whole set. In the first part, I will describe the class of self-similar groups, its important examples, and connections to other classes of groups as well as square tiling problems.
 
In the second talk, I will concentrate on spectacular properties of some self-similar groups: infinite torsion groups, and groups of intermediate word growth. These examples were given by Grigorchuk in the 1980's, and are still the topic of active research. These self-similar groups are "contracting": this is the basis of most proofs, by induction, as well as the source of important algorithmic properties.
 
In the third talk, I will explain a deep and fruitful connection, due to Nekrashevych, between self-similar groups and dynamical systems: there is a bijection between contracting self-similar groups and expanding self-maps of topological spaces. This has led to answers of long-standing open questions in complex dynamics.

Edmund Harriss (University of Arkansas, Fayetteville):

Local rules for aperiodic tilings.

Abstract: 
The discovery of aperiodic sets of tiles, that could tile the plane but not periodically opened up a world in which the local properties of fitting shapes together could enforce long range behaviour. In many of the early examples the behaviour seemed to almost be magic. It was possible to prove the set of tiles was aperiodic, but the proof did not give much insight into why. For hierarchical tilings some general constructions have emerged that allow the creation of local rules for any substitution tiling, though these methods are a little brute force, often giving sets with huge numbers of distinct tiles. 
 
In the first lecture I will describe the history of aperiodic tile sets and prove that the Penrose tiles with matching rules are aperiodic. In the second lecture I will show the general structures on which we might consider local rules, and use these to show the methods that create local rules for and substitution tiling. Finally in the third lecture I will discuss a large family of tilings discovered by Goodman-Strauss that provide a useful space for detailed exploration between the magic and the brute force. I will conclude with some discussion of how these local rules can be applied in many ways from art to mathematics.

Jarkko Kari (University of Turku):

Wang tilings and Cellular Automata: aperiodicity and computability.

   Abstract:

Cellular automata are discrete dynamical systems based on local, synchronous and parallel updates of symbols written on an infinite array of cells. They are the simplest imaginable massively parallel computing systems that operate under the nature inspired constraints of locality of interactions and uniformity in time and space. They can also exhibit the physically relevant properties of time reversibility and conservation laws if the local update rule is chosen appropriately. Closely related notions in symbolic dynamics are tiling spaces: sets of infinite arrays of symbols defined by forbidding the appearance anywhere in the array of some local patterns. In comparison to cellular automata, the dynamic local update function has been replaced by a static local matching relation.

These lectures present classical results about cellular automata, discuss algorithmic questions concerning tiling spaces and relate these questions to decision problems about cellular automata, observing some fundamental differences between the one- and two-dimensional cases.

Lorenzo Sadun (University of Texas at Austin):

Fusion : A general framework for hierarchical tilings.

   Abstract:

Many important classes of tilings exhibit a hierarchical structure. In this mini-course, we'll review the simplest setting, substitution tilings, where the hierarchy looks the same at each level. We will then relax the rules, allowing new structures at different length scales, and see that much (but not all!) of the theory of substitutions carries over with minor modifications. This includes topological constructions, spectral theory, and ergodic theory. We then extend the theory to cover tilings of infinite local complexity, and develop new measures of complexity to understand those examples.

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