Introduction to Louis Michel's lattice geometry through group action

Introduction to Louis Michel’s lattice geometry through group action
	Contents
	Preface
	1 - Introduction
	2 - Group action. Basic definitions and examples
		2.1 The action of a group on itself
		2.2 Group action on vector space
	3 - Delone sets and periodic lattices
		3.1 Delone sets
		3.2 Lattices
		3.3 Sublattices of L
		3.4 Dual lattices
	4 - Lattice symmetry
		4.1 Introduction
		4.2 Point symmetry of lattices
		4.3 Bravais classes
		4.4 Correspondence between Bravais classes and lattice point symmetry groups
		4.5 Symmetry, stratification, and fundamental domains
		4.6 Point symmetry of higher dimensional lattices
	5 - Lattices and their Voronoïand Delone cells
		5.1 Tilings by polytopes: some basic concepts
		5.2 Voronoï cells and Delone polytopes
		5.3 Duality
		5.4 Voronoï and Delone cells of point lattices
		5.5 Classification of corona vectors
	6 - Lattices and positive quadratic forms
		6.1 Introduction
		6.2 Two dimensional quadratic forms and lattices
		6.3 Three dimensional quadratic forms and 3D-lattices
		6.4 Parallelohedra and cells for N-dimensional lattices
		6.5 Partition of the cone of positive-definite quadratic forms
		6.6 Zonotopes and zonohedral families of parallelohedra
		6.7 Graphical visualization of members of the zonohedral family
		6.8 Graphical visualization of non-zonohedral lattices
		6.9 On Voronoï conjecture
	7 - Root systems and root lattices
		7.1 Root systems of lattices and root lattices
		7.2 Lattices of the root systems
		7.3 Low dimensional root lattices
	8 - Comparison of lattice classifications
		8.1 Geometric and arithmetic classes
		8.2 Crystallographic classes
		8.3 Enantiomorphism
		8.4 Time reversal invariance
		8.5 Combining combinatorial and symmetry classification
	9 - Applications
		9.1 Sphere packing, covering, and tiling
		9.2 Regular phases of matter
		9.3 Quasicrystals
		9.4 Lattice defects
		9.5 Lattices in phase space. Dynamical models. Defects
		9.6 Modular group
		9.7 Lattices and Morse theory
	A - Basic notions of group theory with illustrative examples
	B - Graphs, posets, and topological invariants
	C - Notations for point and crystallographic groups
		C.1 Two-dimensional point groups
		C.2 Crystallographic plane and space groups
		C.3 Notation for four-dimensional parallelohedra
	D - Orbit spaces for planecrystallographic groups
	E - Orbit spaces for 3D-irreducible Bravais groups
	Bibliography
	Index