General Report by Konstantin Rybnikov
During the summer of 2001 undergraduate students Michael
Greene (Harvard), Jody Radowicz (North Central), Sarah Crown (Bryn Mawr)
and Kirsten Wikelgren (Harvard) studied geometry of homogeneous and
inhomogeneous quadratic forms over integers. I was helped in this project
by Cornell Ph.D. students Todd Kemp and Franco Saliola. Students worked
in the following directions: geometry and combinatorics of perfect Delaunay
polytopes, connections between extreme L-types and perfect forms, and the
symmetries of the Voronoi polyhedron (Voronoi polyhedron is the convex
hall of all integer points in the cone of positive semidefinitet real matrices).
Keywords: point lattice, quadratic forms over integers,
Delaunay polytope, Delaunay tiling, cone of positive definite matrices,
L-type, Voronoi reduction of the 2nd type, inhomogeneous quadratic form,
extreme hypermetric, diophantian inequality, perfect quadratic form, Voronoi
polyhedron,
Last Update: September 16, 2001