Quadratic forms over number rings

02/23/2010 - 12:15pm
02/23/2010 - 1:10pm
Larry Gerstein (UC Santa Barbara)

The beloved Gram-Schmidt orthogonalization process - the key to classifying inner-product spaces over R - falls short when we consider inner products on modules over rings. For example, if one takes a basis for R^n and generates the linear combinations using only integer coefficients, the result is a Z-lattice L (picture a crystal structure filling R^n), and L need not have any orthogonal decomposition at all. When are two such lattices isometric? What numbers qualify as lengths of vectors in L? These and other issues will be explored for Z-lattices and for lattices over other rings of number-theoretic interest in this expository talk.

Millikan 208 (Pomona College)