Abstracts
Christophe BAVARD (Bordeaux) : Symmetric spaces and euclidean packing theory
Abstract : The study of lattice packings of spheres in the Euclidean
space goes back to Lagrange, Gauss, Hermite and Minkowski.
The talk
will focus on isodual lattice packings. I will explain how the geometry
of symmetric spaces (in particular hyperbolic geometry)
can be
used to solve the maximal density problem in some cases.
Peter BUSER (EPFL) : Löbell manifolds and trigonometry
Abstract : In 1931 Frank Löbell constructed a three dimensional closed hyperbolic manifold by pasting together copies of a
right angled hyperbolic polyhedron that one may view as a three dimensional -albeit rigid- analog of a right angled geodesic hexagon.
The lecture is about joint work with A. Mednykh and A. Vesnin in which we study generalisations of Löbell's construction. The underlying
polyhedra have trigonometric relations that may be proved using hyperbolic isometries in an amusing way.
Karsten GROVE (Notre Dame) : Reflection groups in non-negative curvature
Abstract : We provide an equivariant description / classification of all complete (compact or not) non-negatively curved manifolds M
together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks
consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive
a corresponding splitting theorem for the universal cover. This is joint work with Fuquan Fang.
John RATCLIFFE (Vanderbilt) : Right-angled Coxeter polytopes, hyperbolic 6-manifolds, and a problem of Siegel
Abstract :
In this talk I will survey what is known about Siegel's problem of determining the minimum possible volume of a
complete hyperbolic
n-manifold. In particular I will describe our solution of Siegel's problem for n = 6.
This is joint work with Brent Everitt and Steven Tschantz.
Pavel TUMARKIN (Durham) : Coxeter groups, quiver mutations and hyperbolic manifolds
Abstract : Based on quiver mutations, Barot and Marsh recently constructed a
series of presentations of finite Coxeter groups.
I will discuss a
geometric interpretation of this result: it occurs that these
presentations give rise to a construction of geometric
manifolds with
large symmetry groups, in particular to some hyperbolic manifolds with
proper actions of finite Coxeter groups.
The work (still in progress)
is joint with Anna Felikson.
Ernest VINBERG (Moscow) : Non-arithmetic hyperbolic reflection groups in higher dimensions
Abstract : In 1988, M. Gromov and Piatetski-Shapiro proved that in the hyperbolic space Hn
of any dimension n there exist
non-arithmetic cofinite discrete groups of motions
(both cocompact and non-cocompact). However, non-arithmetic cofinite
reflection
groups in Hn (both cocompact and non-cocompact) were only known for n ≤ 5
by that time. In 1989, O. Ruzmanov
constructed such (non-cocompact) groups for
6 ≤ n ≤10. In the talk, some examples of such groups will be presented for
higher dimensions, including n=18.