Lipschitz mappings, differentiability and exceptional sets.

On an infinite grid sits an electronic circuit consisting of a finite, square number of components connected by wires. The components are positioned in a scattered, irregular formation and each component is connected to every other by wires. Moreover, each wire is unique and can be used to connect only one specific pair of components. Our task is to tidy up this circuit board by assembling the scattered components in a regular square formation of n rows consisting of n components. We first remove all connecting wires and then move the components so that they form a square. Next we have to reconnect the components by reattaching the wires, taking care to attach each wire to the unique pair of components to which it fits. However, after moving the components to a regular square formation the original distances between some pairs of components may have increased, so it is necessary to stretch some of the wires before reattaching. In 2017 it was proved that the initial scattered formation of the circuit board can be so bad that we are forced to stretch some wires by an arbitrarily large factor. This answers a long standing open question of Feige from 2002. However, it remains an open problem to determine for each fixed number n the size of the minimal stretch factor with which any n component scattered circuit board can be successfully reassembled. Moreover, we do not know how the worst possible starting formations of the circuit board, which force us to stretch the wires the most, look like. In this project we seek to answer these questions. Mappings which stretch the distances between points by at most a constant factor are called Lipschitz and form the central object of study of this project. In addition to the problem described above, we investigate special types of Lipschitz mappings on a space which do not squash volume too much. We plan to tackle a long standing open problem of whether such mappings may send infinitely many points to the same image point. Lipschitz functions on finite dimensional spaces are differentiable at almost every point of the space. We seek to answer the question of whether every subset of the plane with zero area can be determined only by looking at the points of differentiability of Lipschitz functions inside of the set. Finally, we study Lipschitz functions on infinite dimensional spaces which do not increase distances between points. The project aims to uncover properties shared by `nearly all` such non-expansive mappings. In particular, we investigate whether it is true that at a given point nearly all non-expansive mappings have the maximal stretch factor one.

In the FWF Project "Lipschitz Mappings, Differentiability and Exceptional Sets", researchers have examined important properties of mappings satisfying a simple stretch condition, namely mappings which are allowed to increase the distance between any two points by at most a constant factor. Mappings satisfying such a condition are called Lipschitz. If this constant factor is equal to one, the stretch condition means that the mappings cannot increase distances between any two points. Such mappings are called nonexpansive. It is possible to define a natural notion of distance between two nonexpansive mappings so that sets of nonexpansive mappings become metric spaces with very nice properties. One theme of the project was the study of properties which are typical in such spaces, meaning that in a certain sense, nearly all nonexpansive mappings possess this property. The project contributed to research in this area, both in the setting of classical finite dimensiona l spaces and in more abstract spaces, where the distance between two points is defined by the length of connecting curves. The stretch condition on Lipschitz mappings brings with it several useful properties. For example, the Lipschitz mappings we consider are almost everywhere differentiable. This means that on very small scales around nearly all points they are very well approximable by much less complicated mappings. There is a small set of points where this local approximation may fail. Thus, Lipschitz mappings are intimately connected with the theory of very small, or exceptional sets. Project researchers uncovered new properties of Lipschitz mappings and their relations with exceptional sets. Notably, they determined that subsets of a finite dimensional space divide into two distinct types, according to the behaviour of Lipschitz functions inside them. A further target of the project was to use Lipschitz mappings as a means of comparing the structures of different metric spaces, according to the following principal: If a large stretch constant is required in order to map one metric space onto another, it indicates that these two metric spaces are rather different. In this work, researchers also considered natural generalisations of Lipschitz mappings whe re the stretch factor is no longer required to be constant, but is allowed to depend on the distance between the two relevant points. This led to a new way of comparing grid like sets in finite dimensional spaces. The results of the FWF Project "Lipschitz Mappings, Differentiability and Exceptional Sets" have interesting implications in the fields of Discrete Metric Spaces, Set Theory and Geometric Measure Theory.