Algebra, Geometry, and Quantisation

Over the coming months, I hope to write a collection of short notes on various algebraic and geometric aspects of quantisation. Here’s a broad outline of the topics I hope to cover.

• Classical Dynamics and Geometry
  • The Action Principle
  • Canonical Structures
  • Symplectic Geometry
• Symmetries
  • The Geometry of Forms
  • Group Actions
  • Hamiltonian Vector Fields
  • Conserved Charges
  • Moment Maps
  • Dual Algebras
• Groups, Algebras, and Representations
• The Heisenberg Algebra
• The Oscillator Formalism
  • Two Dimensions
  • Higher-Dimensional Oscillators
  • Completeness?
  • Three-Dimensional Oscillators, Revisited
  • The Theta Correspondence
  • A Simpler Example: Schur-Weyl Duality
  • Dual Reductive Pairs and Applications
  • Supergroups
• The Hall Effect
  • The Landau Problem
  • Integer Hall Effect
  • Fractional Hall Effect
• Bosonisation
• Geometric Quantisation
  • Topological Effects
  • Coherent States
  • Kähler Manifolds and Quotient Spaces
• Kirillov’s Orbit Method
• Localisation Theorems
• Fermions, Clifford Algebras, and Spinors