Recent Results in the Theory and Application of Quadratic Forms

This talk will mention new results on: the classification of quadratic forms of small determinant, obtained with the help of a new version of the mass formula; the classification of groups of matrices of integer; and finding integral coordinates for integral lattices (generalizing the theorem that every number is a sum of 4 squares). We also describe recent applications of quadratic forms to quantizing (i.e. data compression and to the construction of trellis codes (the latest rage in modems)). (This talk is based on joint work with J.H. Conway and A.R. Calderbank).