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. - 4W1+? 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. Language: English