Nonlinear Geometries and Applications – NoLiGeA

The current project will follow four general directions of research:
Describe and understand the nonlinear geometries of a metric space.
Describe and understand the interaction between the large-scale structure and the small-scale structure of a metric space.
Describe and understand the interplay between the coarse geometry and expansion properties of a metric space.
Uncover new applications in geometric group theory, noncommutative geometry and theoretical computer science.
Carrying out further research in the field covered by the project is motivated by the potential applications in other
disciplines such as quantum physics and theoretical computer science.

Several results have been obtained on the geometry of:
-the Hamming cube
-proper metric spaces
-stable metric spaces
-groups with finite asymptotic dimension.

This project is expected to have a deep and long-lasting impact in coarse metric geometry, in particular in geometric group theory and noncommutative geometry. So far there are very few techniques available to whom wants to coarsely embed groups into Banach spaces. Despite the topic is clearly a metric one in nature, most of the techniques are algebraic and ad-hoc. The project NoLiGeA proposes to reconsider entirely the embedding problem from an innovative and purely metric standpoint. The primary characteristic of the
project is its intersectoriality. The natural introduction of powerful tools from various established fields of mathematics for the study of geometric properties of groups and Banach spaces is expected to make significant advancements towards the research objectives set. In geometric group theory, probabilistic
techniques are far from playing the same predominant role as they have played in Lipschitz geometry. The project NoLiGeA will introduce this concept in a novel form that is tailored to treat fundamental problems in coarse geometry.

F. Baudier, D. Freeman, Th. Schlumprecht, and A. Zsak, The metric geometry of the Hamming cube and applications, submitted for publication, arXiv:1403.4376, 13 pages
F. Baudier and G. Lancien, Optimal embeddability of stable metric spaces, preprint, 12 pages
F. Baudier and M. Ostrovskii, On metric dimension and stochastic decompositions, in preparation.

The definition of a Banach space requires only a few basic notions in Linear Algebra, Functional Analysis and Topology. However the analysis of the geometric properties of the underlying metric space of a Banach space is a highly challenging task. One of the applicant's long-term goal is to give a thorough description of the natural geometries carried by a Banach space, eventually a metric space, and to provide a clear picture of the relationship between them.

It turns out that understanding how well a metric space can be embedded into a particular type of Banach space is a fundamental problem in many diverse fields. For instance embeddings incurring a small distortion on distances, of large but finite metric spaces, into Banach spaces with an almost Euclidean geometry is of high interest to theoretical computer scientists since the 1990s in connection with data-mining. Computer scientists together with Banach space geometers were able to use the significant amount of knowledge in the Lipschitz geometry of Banach spaces which has been studied for itself since the early 1900's. The contribution of this part of the nonlinear geometry of Banach spaces has impacted the design of fast algorithms. Nonlinear problems have been solved using techniques from theoretical computer science as well.

In connection with the Novikov and the Baum-Connes conjectures, there has been an upsurge of interest from geometric group theorists and topologists regarding the coarse geometry of metric spaces, in particular Banach spaces. Unfortunately, unlike the Lipschitz geometry or the uniform geometry, the coarse geometry of Banach spaces had not been considered for itself by Banach space geometers until it became clear, since around 2000, that it had significant applications. One reason is likely to be that coarse maps are not highly structured. They need not be neither injective nor continuous. So far what we understand of the coarse geometry of Banach spaces is mainly derived from what we know of the uniform geometry. One of the project NoLiGeA's main concern is to fill this gap and to carry out the systematic and gigantic but enticing investigation of the nonlinear geometries of fundamental classes of metric spaces.