GEOMETRIC HAMILTON–JACOBI THEORY

Geometric Complexity Theory: Introduction

Foreword These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the...

10 Geometric Graph Theory

In the traditional areas of graph theory (Ramsey theory, extremal graph theory, random graphs, etc.), graphs are regarded as abstract binary relations. The relevant methods are often incapable of providing satisfactory answers to questions arising in geometric applications. Geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edg...

Geometric Semigroup Theory

Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups and automata lead to simplifications of the graphs on which the corresponding finite semigroups act. We show in particular that every finite semigroup can b...

Geometric Singular Perturbation Theory

Geometric Group Theory Notes

Definition. Let Γ be a group. Take A ⊂ Γ to be any subset. The subgroup generated by A, denoted 〈A〉, is the smallest subgroup of Γ containing A. Equivalently, 〈A〉 is the intersection of all subgroups of Γ containing A. Equivalently, 〈A〉 = {a1 1 · · · an n | ai ∈ A, ǫi = ±1}, the set of finite products of elements of A ∪A−1. Notation. We will frequently use Γ to denote a group. If A is a set, A ...