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 1 If 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 2 A 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
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 .
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.