at the

American Institute of Mathematics, Palo Alto, California

organized by

Gabor Pete and Mark Sapir

This workshop, sponsored by AIM and the NSF, will be devoted to percolation on transitive graphs, most importantly, on Cayley graphs of finitely generated infinite groups. Geometric properties of Cayley graphs often turn out to have counterparts in the probabilistic world, and vice versa, but the translations between the different viewpoints are not always trivial. The aim of this workshop is to bring together people working in geometric group theory, probability and dynamics to learn from each other about the relevant techniques in these fields and thus generate new momentum to solve some of the persistent open problems.

Some specific problems we would like address:

- Universality of critical percolation behavior: the conjectures regarding
*p*<1,_{c}*p*<_{c}*p*,_{u}*p*<1, quasi-isometry invariance, critical exponents, the role of asymptotic cones. Could scaling limits be described analogously to SLE?_{u} - What is the relation between lace expansion and the triangle condition of percolation theory, the rapid decay property, and hyperbolicity of groups?
- On what groups is renormalization possible?
- Relation of group theoretic properties (Kazhdan's property T, L
^{2}and bounded cohomology, cost of groups, finite presentability) to percolation. - Survival of geometric and random walk properties under percolation.
- The role of unimodularity in percolation.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*

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