L1

Two dimensional minimal cones were fully classified by Jean Taylor in the mid

70's.  In joint work with G. David and T. De Pauw we prove that a closed

set which is close to a minimal cone at all scales and at all locations is

locally a bi-Hoelder image of a minimal cone.  This result is analogous to

Reifenberg's disk theorem.  A couple of applications will be discussed.

My lecture is based on results of [1] and [2]. In [1] we use an extension of the method due to Borel and Prasad to determine the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group. In [2] the results of [1] are combined with the previously known asymptotic of the number of subgroups in a given lattice in order to study the general lattice growth. We show that for many high-rank simple Lie groups (and conjecturally for all) the rate of growth of lattices of covolume at most $x$ is like $x^{\log x}$ and not $x^{\log x/ \log\log x}$ as it was conjectured before. We also prove that the

conjecture is still true (again for "most" groups) if one restricts to counting non-uniform lattices. A crucial ingredient of the argument in [2] is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.

I plan to give an overview of these recent results and discuss some ideas beyond the proofs.

[1] M. Belolipetsky (with an appendix by J. Ellenberg and A.

Venkatesh), Counting maximal arithmetic subgroups, arXiv:

math.GR/0501198.

[2] M. Belolipetsky, A. Lubotzky, Class field towers and subgroup

growth, work in progress.

A coupled cell system is a collection of interacting dynamical systems.
Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much depends on the specific equations?

The ideas will be illustrated through a series of examples and theorems. One theorem classifies spatio-temporal symmetries of periodic solutions and a second gives necessary and sufficient conditions for synchrony in terms of network architecture.