Title: Combinatorics of orbits in the space of lattices near the cusp

Abstract: Consider the space of rank n lattices with covolume 1. This space is not compact and has a cusp. We study the action of a one-parameter diagonal group on the space of lattices, and specifically ask: how do orbits behave near the cusp? We introduce certain combinatorial patterns that describe the behavior of orbits near the cusp. We discuss why every such pattern can be realized (thereby showing that our description is complete), and how many orbits (measured in terms of Hausdorff dimension) exhibit each pattern.