1 Recent Developments in Affine Toda Quantum Field Theory.- 1 Introduction.- 2 Classical Integrability and Classical Data.- 3 Aspects of the Quantum Field Theory.- 4 Dual Pairs.- 5 A Word on Solitons.- 6 Other Matters.- 7 References.- 2 A Class of Fermi Liquids.- 1 Introduction.- 2 Four-Legged Diagrams.- 3 A Single-Slice Fermionic Cluster Expansion.- 4 References.- 3 Quantum Groups from Path Integrals.- 1 Classical Field Theory.- 2 Categories, Finite Groups, and Covering Spaces.- 3 Generalized Path Integrals.- 4 The Quantum Group.- 5 References.- 4 Half Transfer Matrices in Solvable Lattice Models.- 1 The Six-Vertex Model.- 2 The Antiferromagnetic Regime.- 3 Corner Transfer Matrix.- 4 Half Transfer Matrix.- 5 Commutation Relations.- 6 Correlation Functions.- 7 Two-Point Functions.- 8 Discussion.- 9 References.- 5 Matrix Models as Integrable Systems.- 1 Introduction.- 2 The Basic Example: Discrete 1-Matrix Model.- 3 Generalized Kontsevich Model.- 4 Kp/Toda ?-Function in Terms of Free Fermions.- 5 ?-Function as a Group-Theoretical Quantity.- 6 Conclusion.- 7 References.- 6 Localization, Equivariant Cohomology, and Integration Formulas 211.- 1 Symplectic Geometry.- 2 Equivariant Cohomology.- 3 Duistermaat-Heckman Integration Formula.- 4 Degeneracies.- 5 Equivariant Characteristic Classes.- 6 Loop Space.- 7 Example: Atiyah-Singer Index Theorem.- 8 Duistermaat-Heckman in Loop Space.- 9 General Integrable Models.- 10 Mathai-Quillen Formalism.- 11 Short Review of Morse Theory.- 12 Equivariant Mathai-Quillen Formalism.- 13 Equivariant Morse Theory.- 14 Loop Space and Morse Theory.- 15 Loop Space and Equivariant Morse Theory.- 16 Poincaré Supersymmetry and Equivariant Cohomology..- 17 References.- 7 Systems of Calogero-Moser Type.- 1 Introduction.- 2 Classical NonrelativisticCalogero-Moser and Toda Systems.- 3 Relativistic Versions at the Classical Level.- 4 Quantum Calogero-Moser and Toda Systems.- 5 Action-Angle Transforms.- 6 Eigenfunction Transforms.- 7 References.- 8 Discrete Gauge Theories.- 1 Broken Symmetry Revisited.- 2 Basics.- 3 Algebraic Structure.- 4 $${\overline D _2}$$ Gauge Theory.- 5 Concluding Remarks and Outlook.- 6 References.- 9 Quantum Hall Fluids as W1+? Minimal Models.- 1 Introduction.- 2 Dynamical Symmetry and Kinematics of Incompressible Fluids.- 3 Existing Theories of Edge Excitations and Experiments.- 4 W1+? Minimal Models.- 5 Further Developments.- 6 References.- 10 On the Spectral Theory of Quantum Vertex Operators 469 Pavel I. Etingof.- 1 Basic Definitions.- 2 Spectral Properties of Vertex Operators.- 3 The Semi-Infinite Tensor Product Construction.- 4 Computation of the Leading Eigenvalue and Eigenvector.- 5 References.