||Time and place||Thursday 10:10-12 Schreiber 210
||About||We will amuse ourselves by reading some papers about applications of homogeneous dynamics in number theory (particularly diophantine approximation and geometry of numbers) and geometry of discrete sets (Delone sets).||
Tilings, quasicrystals, discrete planes, generalized substitutions and multidimensional continued fractions
following Arnoux, Berthe, Ei and Ito.
Here is the relevant paper.
Obvious divergence for cones
Many diophantine properties have a dynamical interpretation. We will discuss the one concerning divergence on cones, and see what stands behind the definition of the obvious ones. We will show that under certain cones of SO(2,2) all divergent trajectories are the obvious ones.
A problem of counting lattice points in the hyperbolic plane
We present an elegant result by M. Risager and Z. Rudnick about the statistics of the shortest solution to the Diophantine equation ax-by=1, where a and b are co-prime ("the gcd equation"). The proof is essentially carried through counting lattice points in a family of expanding domains in the hyperbolic plane. We present a result about effective lattice point count in these domains, and in analogous domains in the hyperbolic n-space for all n>1. In particular, we conclude a generalization of Risager and Rudnick's result for rings of integers in quadratic imaginary number fields. This is a joint work with Amos Nevo.
This will be a review of basic properties of Pisot numbers. A good source is the last chapter of Cassels' book, An Introduction to Diophantine Approximation. I will follow the paper of Jaroslaw Kwapisz, A dynamical proof of Pisot's theorem.
Stationary measures on the space of rank 2 discrete subgroups of 3-dimensional space.
The space of rank 2 discrete subgroups of R^3 can be realised as a homogeneous space G/H. However H is not a lattice or even a discrete subgroup of G. Using recent developments of Y. Benoist and J.F. Quint we are able to analyse stationary measures for Zariski dense subgroups on this space. This work is joint with Uri Shapira.
Suggested reading: sections 3 and 4, and specifically section 4.3, of the following survey of Benoist and Quint.
Towards quasicrystals in non-commutative spaces
In 2011, the Technion chemist Dan Shechtman was awarded the nobel prize for discovering the physical existence of solids with a quasi-crystalline structure by laser experiments (diffraction experiments).
Some years later, certain mathematical physics communities became interested in describing the appearing phemomena precisely and rigorously by a specific class of models called cut-and-project schemes. Most of the literature on this topic deals with crystalline and quasi-crystal line structures in abelian groups. Within the scope of a project with Michael Bjorklund and Tobias Hartnick, we have started to investigate these models in non-commutative (including non-amenable) situations. We start the talk with a brief introduction of abelian cut-and-project schemes, illustrating them by the Fibonacci diffraction model. In a second part of the talk, we shed some light on our progresson a diffraction setup for non-abelian groups.
Note: this is very much work in progress.
THE TALK WILL TAKE PLACE ON THURSDAY AT 11:10
(Almost) Everything Is Illuminated
A translation surface, roughly, is a compact Riemann surface together with an atlas covering all but a finite number of points ("singularities"), where the transition functions are translations. Geometrically, a translation surface can be represented by a certain polygon with sides identified. Following [LMW], I will show, via analysis of an SL(2,R) action on the moduli space of all translation surfaces, that any point on a translation surface has at most finitely many other points which it does not illuminate.
Suggested Reading: (In this order)
- Wikipedia: https://en.wikipedia.org/wiki/Illumination_problem
- Sections 1 and 3 of Zorich, A. (2006). Flat Surfaces. In Frontiers in number theory, physics, and geometry
- [LMW] LELIEVRE, S., MONTEIL, T., & WEISS, B. (2014). Everything is illuminated. Preprint, Available at http://arxiv.org/abs/1407.2975437