After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit­ ful again. Many of the leading specialists of both fields became involved in this devel­ opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im­ portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me­ chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, ·we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth­ while.

Springer Book Archives

to the 18th Conference on 'Differential Geometrical Methods in Theoretical Physics'.- Baxterization.- Geometric Classification of Commutative Algebras of Ordinary Differential Operators.- Geometrical Aspects of Solvable Two Dimensional Models.- Explicit Soliton-Generating Bäcklund Transformations.- Integrability Conditions: Recent results in the theory of integrable models.- Nonlinear Differential Equations in Physics and Their Geometrical Integrability Properties.- Integrability Off Criticality and Quantum Integrable Systems.- Quantization of the Chiral Solitonic Bag Model.- Structure of Superselection Sectors in Low-Dimensional Quantum Field Theory.- Cyclic Cohomology, Supersymmetry and KMS States the KMS States as Generalized Elliptic Operators.- Symmetrics of Quantum Space, Braid Representation, and Classification of Subfactors.- New Kinematics (Statistics and Symmetry) in Low-Dimensional QFT with Applications to Conformal QFT2.- Infinite Index Embeddings.- Non-Compact Current Algebras and Heterotic Superstring Vacua.- Aspects of Perturbed Conformal Field Theory, Affine Toda Field Theory and Exact S-Matrices.- Codes, Lattices and Conformal Field Theory.- Topics on Conformal Field Theory.- Feigin-Fuchs Representation of Conformal Field Theory.- Coulomb-Gas Construction on Higher-Genus Riemann Surfaces.- Conformal Algebras and Non-linear Differential Equations.- S Matrices of the Tricritical Ising Model and Toda Systems.- Quantum Groups, Braiding Matrices and Coset Models.- Gauged WZW Models and the Coset Construction of Conformal Field Theories.- Flat Connection, Conformal Field Theory and Quantum Group.- Classical and Quantum Calabi-Yau Manifolds.- Conformal Field Theories and Category Theory.- Quantum Bäcklund Transformations and Conformal Algebras.- ACoset-Construction for Integrable Hierarchies.- Away From Criticality: Some Results From the S Matrix Approach.- Normal Ordered Products and Parafields in Conformal Field Theory.- Chiral Gauge Field Theory in Two Dimensions.- Monodromy Properties of Conformal Field Theories and Quantum Groups.- Matrix elements of unitary representations of the quantum group SUq(1, 1) and the basic hypergeometric functions.- A q-Analogue of the Lie Superalgebra OSp(2,1) and its Metaplectic Representation.- q-Deformation of SU(1, 1) Conformal Ward Identities and q-Strings.- Q-Deformation of sl(2, c) × ZN and Link Invariants.- Quantum Group Duality in Vertex Models and Other Results in the Theory of Quasitriangular Hopf Algebras.- Physics at the Planck Length and p-Adic Field Theories.- Non-Archimedean Geometry and Applications to Particle Theory.- Beyond Conformal Field Theory.- Hidden Symmetries of Strings and Their Relevance for String Quantization.- Hamiltonian Flows, SU(?), SO(?), USp (?), and Strings.- A Geometric Approach to the String BRS Cohomology.- Progress in Multi-Genus Calculations for the Spinning String.- The Minimal Set of the Generators of Dehn Twists on a High Genus Riemann Surface.- Strings and Teichmueller Space.- Holomorphic Differentials on Punctured Riemann Surfaces.- Anomalies, BRS Symmetry and Superconnections.- General Covariance and Strings.- Spontaneous Symmetry Breaking in 4-Dimensional Heterotic String.- Superghost Fields in N = 2 Superconformal Algebra.- Topological Quantum Field Theories: Relations Between Knot Theory and Four Manifold Theory.- Topological Quantum Theories and Representation Theory.- Topological Chern-Simons Gauge Theories and "New" Knot/Link Polynomials.- Observables in Topological Yang-Mills Theory and the Gribov Problem.-Linking the Gauss-Bonnet-Chern Theorem, Essential Hopf Maps and Membrane Solitons with Exotic Spin and Statistics.- Moduli Spaces and Topological Quantum Field Theories.- Knots in Physics.- Supermanifold, Symplectic Structure and Geometric Quantization of BRST Systems.- Ambitwistors and Conformal Gravity.- Toward Classification of Classical Lie Superalgebras.- Status of the Algebraic Approach to Super Riemann Surfaces.- Projective Embeddings of Complex Supermanifolds.- Some Results on Line Bundles over SUSY-Curves.- Instantons From Supersymmetric Conformal Chiral Scalar Superfield Theories.- Einstein-Hermitian Bundles over Complex Surfaces.- Symplectic Reduction of the Minimally Coupled Massless Superparticle in D=10.- Quantum Gravity and the Berry Phase.- Gravity and Lorentz Breakdown in Higher-Dimensional Theories and Strings.- Current Algebra and Extended 2D Gravity with Higher Spin Gauge Field.- The Parametric Manifold Picture of Space-Time.- Gravity as an SO(3, 2) Gauge Theory.- Heuristics of Solitary Waves in Non-Integrable Field Theories.- Lattice Approach of the Antiferromagentic Heisenberg Model in 2+1 Dimensions and the Hopf Chern-Simons Terms.- Is It Possible To Do Canonical Quantum Field Theory Rigorously?.- A Global Theory of Parametrized Quantum Mechanics.- Principal Bundles Versus Lie Groupoids in Gauge Theory.- Static and Axially Symmetric Soliton Solutions to the Self-Dual SU(3) and SU(2) Gauge Fields in a Euclidean Space.- Superalgebra and Superspace of Vector Spinor Generators.- Participants.- Author Index.