Despite widespread interest in cryptographic multilinear maps since
Boneh-Silverberg's 2003 paper, very few candidate maps have been
discovered. The first serious candidate was a scheme of
Garg-Gentry-Halevi (GGH), which is based on ideal lattices in cyclotomic
number rings. While the scheme was later shown to be broken, the only
other candidate schemes are hardened variants of GGH. We give a
relatively detailed description of the GGH multilinear map.

We consider random products of SL(2, R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice. This is a joint project with A.Gorodetski.

In '99, David Aldous conjectured that a certain natural "random walk" on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.