Group geometry and the minima of Hermitian forms
Eduardo Mendoza (Max Planck Institute of Biochemistry, Munich, Germany)
1 PM, Monday, 6 February 2012, Room 117
 
Abstact: Computing the minima of quadratic forms (on lattices over integers in algebraic number fields) is a classical problem in number theory and is closely related to optimal sphere packings. A wide variety of algorithms have been developed in this regards, particularly for lattices over Z. 
 
In this talk we consider the case of binary Hermitian forms and discuss recent results which use the geometry of the Mendoza complex of the associated Bianchi group for the computation. For positive definite forms, Chan, et al introduce the concept of a projective Hermite constant which can be computed via the results of Mendoza and Vogtmann and oftern coincides with he (quite inaccessible) classical one.For indefinite forms, Bestvina and Savin introduce the concept of an ocean in the Mendoza complex, generalizing Conway's 'river' in Serre's tree associated with SL(2,Z) and use it to derive interesting characteristics of the forms, in particular their minima. Both papers demonstrate the power of a 'geometric group theory' approach to mathematical problems.
 

Tiling space by translates of a convex body, with multiplicity
Sinai Robbins (Nanyang Technological University, Singapore)
1 PM, Monday, 26 September 2011, Institute of Mathematics Conference Room
 
Abstract:   We review some of the history of tilings of space by translations, and generalize this theory to include the problem of covering$ \mathbb{R}^d $by overlapping translates of a convex body P, such that almost every point of$ \mathbb{R}^d $is covered exactly k times, for a fixed integer k. Such a covering of Euclidean space by translations is called a k-tiling. The traditional investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski.   Here we extend the investigations of Fedorov and Minkowski to k tilings by proving that if a convex body k-tiles$ \mathbb{R}^d $by translations, then it is centrally symmetric, and its facets are also centrally symmetric. The methods are very new, and they allow us to prove analogues of Minkowski’s conditions for 1-tiling polytopes. Conversely, in the case that P is a rational polytope, we also prove that if P is centrally symmetric and has centrally symmetric facets, then P must k-tile$ \mathbb{R}^d $for some positive integer k.  We highlight the combinatorial approach, but there is also an Harmonic analysis approach, which we point out.
 
This research is joint work with Nick Gravin and Dmitry Shiryaev.