Title: Coarse embeddings into Banach spaces

Abstract. The notion of a coarse embedding of a metric space into another is very weak form of embedding which can be thought of an embedding which `preserves' the geometry in very large scale. This notion became a central tool in several areas of mathematics; Geometric Group Theory, Topology, Metric Geometry, and even Computer Science. In each of these, roughly the program is to coarsely embed the metric space in question into a nice Banach space and solve the problem there. Usage of coarse embeddings of uniformly discrete metric spaces into sufficiently good Banach spaces in topology was initiated by Gromov. A big breakthrough in Gromov's program was achieved by Yu who showed that if a finitely generated group $G$ admits a coarse embed- ding into some uniformly convex Banach space, then $G$ satisfies the Novikov conjecture and the coarse Baum-Connes conjecture. This and other related developments naturally brought the question of coarse embedding in Banach spaces, especially finding linear obstructions to such an embedding. We will give a brief overview of the state of the art of the problem, and plan to give proofs of two fundamental questions that started it all. 1. Does every separable Banach space coarsely embed into a Hilbert (reflexive) space? (No, Enflo, Kalton) 2. Does (separable) Hilbert space coarsely embed into every other separable Banach space? (No, Baudier, Lancien, Schlumprecht)

Based on joint work with Steve Jackson and Cory Krause.