Proposed by M. Gromov

1. (1) Look at coarse Lipschitz maps from uniformly contractible spaces or contractible coverings of compact manifolds to euclidean space. One wants them to have nonzero degree, but we don't want them to collapse things too much. Translate to the language of C*-algebras. When do you have maps to euclidean space? Points which are far away to begin with should stay far away.

2. (2) Can one compute (or express) something like Lp-cohomology by C*-techniques or C*-invariants?

3. (3) Does an amenable discrete group admit a proper isometric action on a Hilbert space (in the metric sense of proper)? A suitable generalization of the notion of almost flat bundles might yield a proof of the Novikov Conjecture for amenable groups.

4. (4) Does every finitely generated or finitely presented group admit a uniformly metrically proper Lipschitz embedding into a Hilbert space? Even such an embedding into a reflexive uniformly convex Banach space would be interesting. This seems hard.

5. (5) Can one give a new proof using the above philosophy (of mapping to Euclidean space or Hilbert space) of the Strong Novikov conjecture (injectivity of the assembly map for the K-theory of the group C*-algebra) or the Baum-Connes Conjecture for discrete subgroups of SO(n, 1), SU(n, 1).

6. (6) Is there a discrete group г (other than ℤ) such that Bг is finite, and such that some compactification of Eг, satisfying the “compact sets become small at infinity” condition, maps to a compactification of ℝ? A homological version of this would be to ask if there is a finite Bг so that the coarse cohomology of г is zero in all dimensions.

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