**1. Itai Benjamini **

Take Gromov’s density model for random groups. Fix generators. Pick independently and uniformly at random relators of length . Gromov shows that, when is large, the quotient is infinite if . Zuk shows that the quotient has property (T) if .

As Linial explained to us for the Erdös-Renyi model, let us pick relators one after the other. What is the girth of the last infinite quotient ? Growth will collapse, but Girth should behave continuously.

**2. Valerio Capraro **

With Marco Scarsini, we study relations between game theory and amenable groups.

Recall that a countable group is amenable iff it admits bi-invariant means (i.e. finitely additive probability measures).

Here is a 2-player game: Fix a subset . P1 chooses . P2 chooses . P1 and P2 exchange 1 euro wether or not.

Does the game admit a Nash equilibrium ?

Our intuition is that, without further knowledge, players play “casually”, a notion which makes sens for amenable groups only.

Theorem 1If is amenable and if every left-invariant measure is also right-invariant (and conversely), then for all , there are Nash equilibria given by bi-invariant means.

**Question**. Variational problem in amenable groups. Fix a left-invariant mean . Does the following functional on means attain its maximal ?

Beware that Fubini does not hold for means. Also, the functional is not weakly continuous.

**Question**. Game theory characterization of amenable groups. Is it true that a countable group is amenable if and only if for all , there are Nash equilibria ?

**3. Ryokichi Tanaka **

Let be hyperbolic plane. Consider a circle packing, and its contact graph. Perform simple random walk on it. When it is transient, what is the hitting distribution on the boundary circle ? Is it singular with respect to Lebesgue measure ?

When the packing is invariant under a Fuchsian group, harmonic measure may be singular.

**4. Shalom Eliahou **

I start with a hard and ancient problem, the Hadamard conjecture.

Definition 2A Hadamard matrix is a square matrix with entries , which is orthogonal up to factor .

In 1893, Jacques Hadamard conjectured that Hadamard matrices exist for all sizes . The least open case is 668.

I am interested in a weaker form, where one merely requires that mod . This was introduced by Marrero and Butson in 1972. They give a positive answer for and . This is easy.

In 2001, with Michel Kervaire, I gave a positive answer for . It is harder: we provide a list which is a mixture of successful guesses, algebra, number theory. What about ? I hope it is

**5. Nati Linial **

** 5.1. Continuation **

Here is an other relaxation of Hadamard’s conjecture.

What is the least for which it can be shown that there exists matrices which each inner product is between .

** 5.2. Signings **

Let be a -regular graph. A signing of is a symmetric matrix that is obtained by putting ‘s on the entries of ‘s adjacency matrix.

**Conjecture**. Every -regular graph has a signing with spectral radius .

If it holds, then there exist Ramanujan graphs of arbitrarily large size. Indeed, a signing is a specification for a double cover of , and the spectrum of resflects the new eigenvalues occuring for that graph. If was Ramanujan, so is the cover.

** 5.3. Improving on the Moore bound **

Easy: A -regular -vertex graph has girth at most .

**Conjecture**. The factor of is not optimal.

Moore’s bound is an equality for finitely many graphs.

Random graphs have short cycles. Solving the conjecture for Cayley graphs would be interesting.