K. Rybnikov

U. Mass. Lowell

Northeastern University

Monday, January 28, 2008

Talk at 1:30 PM in 511 Lake

Abstract: I will start from discussing the classical problem of finding the densest packings of spheres. This problem is extremely hard even in dimension 3, so I will digress to the special case where the centers of the spheres form a lattice, i.e. the set of integer linear combinations of some n linearly independent vectors. Finding the densest lattice packings of spheres in dimension n is the same as finding positive-definite quadratic forms in n variables which have the highest value of Hermit invariant among all such forms in n variables (in case you prefer the language of number theory).

One of possible approaches to this problem is to consider the space of parameters for congruence classes of n-dimensional lattices. This space of parameters turns out to be the cone of positive quadratic forms in n variables. This cone also happens to be a natural environment for observing the behavior of SL(n,Z), the group of integer matrices of determinant 1.

Arithmetic, geometry, groups, and topology naturally come together in the study of the cone of positive quadratic forms and SL(n,Z). Most of the talk will be of expository nature and will not require any background beyond linear algebra and multivariate calculus. However, I will also touch upon more advanced topics, such as algorithms in geometry of numbers and cohomology of SL(n,Z).

Here are some directions to Northeastern University. Lake Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street.

GASC Seminar Home Page	Posted: January 23, 2008.
Web page: Alexandru I. Suciu	URL: http://www.math.neu.edu/gasc/abs/rybnikov08.html