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Analysis, geometry and group theory are three of the main classical areas of mathematics. Analysis studies local properties of a space, geometry is concerned with its overall structure, while group theory wants to know the space's symmetries. A metric space is an example of an object which is potentially of interest to all three, and this proposal is concerned with uniformly discrete metric spaces. Such spaces, where the distance between any two points is never smaller than some fixed number (think of stars on a dark night) do not seem to have enough local structure to make them interesting from the point of view of analysis, but just as a collection of stars begins to display an intricate shape if we look at it from a large distance (this is how we observe galaxies), a discrete metric space becomes an interesting analytic object if we study it on the large scale. Gromov and Roe provided a concrete scheme for using this simple insight; the result, coarse geometry, is now an established tool, and has been particularly successful when applied to discrete groups. In a finitely generated group, where every element can be written as a word using a finite alphabet, short words can be regarded as being close to the identity element, and long words far away from it. This leads to a natural metric, which gives the group a shape that is homogeneous (looks the same from every point) and symmetric (the whole group provides symmetries of this space). This space can be described analytically through the properties of the reduced C*-algebra, and some of the most important questions in this area of mathematics, like the Baum-Connes conjecture, arise from desire to understand the structure of this algebra. This proposal arises from our discovery that metric spaces which locally resemble groups in the coarse-geometric sense share with groups a lot of interesting analytic properties. Moreover, we have developed an invariant that allows us to say when a metric space is sufficiently similar to a group. Our main new idea, the partial translation structure on a metric space, captures the key combinatorial properties of the left and right mulitplication action of a group on itself and provides a method of encoding those properties in a new C*-algebra, the partial translation algebra, that we associate with a metric space. A unifying strategy of this proposal is the development of partial translation structures, partial translation C*-algebras and our invariant to provide new routes of attack on important outstanding problems. A difficult and much studied question is when a metric space admits a uniform embedding into a Hilbert space or a group. Such an embedding allows one to control the large scale geometry of a space: we compare the space with an object of known geometry, in the first case, or known symmetry, in the second. We will further develop our techniques to construct new counterexamples to the coarse Baum-Connes conjecture, which is an important organising principle for a large body of research on the interface between the analysis and geometry of groups and metric spaces. Our approach will also provide insights into the Valette conjecture, which is an important open question in geometric group theory. This proposal is timely, ambitious and demanding, and is placed in an exciting, rapidly developing and competitive area of mathematics.
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