The shape of sublattices of Z^m

04/17/2012 - 12:15pm
04/17/2012 - 1:10pm
Wolfgang Schmidt (University of Colorado at Boulder)

A lattice in R^m is a discrete subgroup. It is a free abelian group of some rank n not exceeding m. Being embedded in Euclidean space R^m, it has some "shape". Lattices will be considered to have the same shape if they are "similar", i.e. if one can be obtained from the other be an angle preserving linear map. Given a set D of shapes, i.e. of similarity classes, what proportion of lattices has shape in D ? In particular, how many sublattices of Z^m, and with "determinant" at most T, have shape in D ? I will present old and new results, give generalizations, and discuss how to obtain good error terms.

Millikan 208 (Pomona College)
Misc. Information: 

Professor Schmidt will also give a Colloquium talk on Wednesday, 4/18/12.