In the domain of quantum gravity, people often consider random metrics on surfaces, defined as Riemannian ones with the factor being an exponent of the Gaussian Free Field. Though, GFF is only a distribution, not a function, and its exponent is not well-defined. This leaves open the problem of giving a mathematical sense to this definition (or, to be more precise, of showing rigorously that one of the known regularization procedures indeed converges).

In a joint work with M. Khristoforov and M. Triestino, we approach a « baby version » of this problem, constructing random metrics on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.). This situation is still accessible due to the graph structure, but already shares with the planar case the complexity of high non-uniqueness of candidates for the geodesic lines. The behavior of some (pivotal, bridge-type) graphs seems to be a good model for the behavior of the full planar case.

Our eventual goal is to see that if X is any linear order that isomorphic to its cube, then X^{\omega} has a parity-reversing automorphism. Then by the results of last week, X will also be isomorphic to its square. This week, I will describe a method for building partial parity-reversing automorphisms on any A^{\omega}, and give structural conditions on A under which these partial automorphisms can be stitched together to get a full p.r.a. We will see in particular that if A is countable, then A^{\omega} has a p.r.a.

In this talk I will survey some of the cryptosystems based on group theoretic problems and their computational complexity such as Conjugacy, Membership, Endomorphism, Word, Twisted Conjugacy, and Geodesic Length Problems. I will speak about some non-abelian groups that have been proposed as platforms for such cryptosystems: Braid, Polycyclic, Metabelian, Grigorchuk, Thompson, Matrix, Hyperbolic, Small Cancellation, right angled Artin Groups and free nilpotent p-groups. The focus of the talk will be on infinite polycyclic group-based cryptosystems as well as a cryptosystem based on semidirect product of (semi)-groups. The latter is a joint work with V. Shpilrain. There will be open problems related to both computational complexity of group theoretic problems and cryptographic problems that I will mention at the end of the talk. The talk will be accessible to graduate students in computer science and mathematics.

Consider a random product of i.i.d. matrices, randomly chosen from SL(2,R), satisfying some reasonable nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric).Then a classical Furstenberg theorem implies that the norm of such a random product almost surely grows exponentially.

Now, what happens if each of these matrices depends on an additional parameter? We will discuss the case when the dependence of angle is monotonous w.r.t. the parameter: increasing the parameter «rotates all the directions clockwise».

It turns out that (under some reasonable conditions)
- Almost surely for all the parameter values, except for a zero Hausdorff dimension (random) set, the Lyapunov exponent exists and equals to the Furstenberg one.
- Almost surely for all the parameter values the upper Lyapunov exponent equals to the Furstenberg one
- At the same time, in the no-uniform-hyperbolicity parameter region there exists a dense subset of parameters, for each of which the lower Lyapunov exponent takes any fixed value between 0 and the Furstenberg exponent.

The talk is based on a joint project with A. Gorodetski.

We continue toward a proof of the main theorem by characterizing, for a fixed linear order A, the collection of linear orders X such that AX is isomorphic to X, finishing the argument we started with the last time.