The book describes the close relationship between physics and geometry as it is understood nowadays. The reader will find inportant contributions to non-commutative geometry, quantum groups, conformal field theory, and integrable systems.

Higgs fields and superconnections.- Noncommutative differential geometry, quantum mechanics and gauge theory.- to non-commutative geometry and Yang-Mills model-building.- II. Gauge-field model-building via non-commutative differential geometry.- Measuring coalgebras, quantum group-like objects, and non-commutative geometry.- Tensor Operator Structures in Quantum Unitary Groups.- Quantum groups and quantum complete integrability: Theory and experiment.- Some ideas and results on integrable nonlinear evolution systems.- An algebraic characterization of complete integrability for Hamiltonian systems.- Integrable lattice models and their scaling limits QFT and CFT.- Quantum groups, Riemann surfaces and conformal field theory.- Some physical applications of category theory.- From poisson groupoids to quantum groupoids and back.- Quantization on Kähler manifolds.- A new class of infinite-dimensional Lie algebras (continuum Lie algebras) and associated nonlinear systems.- Exchange Algebra in the Conformal Affine sl 2 Toda Field Theory.- Some properties of p-lines.- Breaking of supersymmetry through anomalies in composite spinor operators.- Conformal field theory and moduli spaces of vector bundles over variable Riemann surfaces.- Instanton homology.- W- geometry.- Connections between CFT and topology via Knot theory.- Stochastic calculus in superspace and supersymmetric Hamiltonians.- Geometric models and the modulli spaces for string theories.- Supersymmetric products of SUSY-curves °.- Classical superspaces and related structures.- Remarks on the differential identities in Schouten-Nijenhuis algebra.- Generic irreducible representations of classical Lie superalgebras.- Krichever construction of solutions to the super KP hierarchies.- The structure of supersymplecticsupermanifolds.- Gauge fixing: Geometric and probabilistic aspects of yang-mills gauge theories.- A renormalizable theory of quantum gravity.- Third order nonlinear Hamiltonian systems: Some remarks on the the action-angle transformation.- Tensor products of q p = 1 quantum groups and WZW fusion rules.- The modular group and super-KMS functionals.- New quantum representation for gravity and Yang-Mills theory.- Geometric quantization of the five-dimensional Kepler problem.- Structure functions on the usual and exotic symplectic and periplectic supermanifolds.- Symbols alias generating functionals - a supergeometric point of view.- Sheaves of graded Lie algebras over variable Riemann surfaces and a paired Weil-Petersson inner product.