My talk will be about two open questions and a (perhaps surprising) link between them:

(1) Connes' rigidity conjecture, that in particular predicts that the von Neumann algebras of PSL_n(Z) are not isomorphic for different values of n. Ancient works with Vincent Lafforgue and Tim de Laat suggest a possible approach to it: does the non-commutative Lp space of the von Neumann of SL(n,Z) have the completely bounded approximation property for some non-trivial p?

(2) Kakeya conjecture : every subset of R^d containing a unit segment in every direction has dimension d.Both questions are open for large values of the parameters (n>2 and >3). I will explain why (1) is difficult: it implies some form of (2) for d<=(n+1)/2.