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.
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.- A
Coset-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.
