CONTENTS -- Freydoon Mansouri (1936-2003) -- Preface -- Opening Remarks H.-D. Doebner -- I. Freydoon Mansouri Memorial Lectures -- Some Consequences of the Hypothesis of Minimal Lengths L. N . Chang -- Acknowledgments -- References -- Strings and Black Holes: An Emergence of Patterns? G. Domokos and S. Kovesi-Domokos -- 1. Introduction -- 2. Equivalence of the String and Black Hole Pictures -- 3. Discussion -- References -- Physics and Geometry P. G. 0. fieund -- References -- Neutrinos: Windows to Planck Physics P. Ramond -- 1. A few words about Freyd -- 2. Neutrino data -- 3. Mixings in the Standard Model -- 4. Hints from Grand Unification -- 5.The origin of large angles -- Acknowledgments -- References -- Hamiltonians of Odd Fermion Number and Path Integrals C. M. Sommerfield -- 1. Fred Mansouri -- 2. Introduction and motivation -- 3. The vacuum state of supersymmetric matrix models -- 4. Eigenstates of non-hermitian fermionic operators as basis states -- 5. Fermionic path integral -- 6. Infinitesimal time steps as Grassmann numbers -- 7. Conclusions -- References -- II. Algebras and Representations -- DAHA and Verlinde Algebras I. Cherednik -- 1. Origins of DAHA -- 2. From Lie Theory to DAHA -- 3. Double Hecke Algebra -- 4. Polynomial Representation -- 5. Nonsymmetric Verlinde Algebras -- 6. Deformation -- 7. Rational Degeneration -- 8. Gaussian Sums -- 9. Diagonal Coinvariants -- Contractions and Deformations of Lie Algebras M. d e Montigny -- 1. Introduction -- 2. Contractions of Lie algebras -- 3. Graded contractions of Lie algebras -- 4. Deformations of Lie algebras -- Acknowledgments -- References -- On the Unitarity of D = 9,10,11 Conformal Supersymmetry V. K . Dobrev, R. B. Zhang and B. S. Zlatev -- 1. Introduction -- 2. UIRs of the superalgebras osp(ll2n, R) -- 3. Unitarity for the conformal subalgebras -- References.

Realizations of Exceptional U-Duality Groups as Conformal and Quasi-Conformal Groups and Their Minimal Unitary Representations M. Gunaydin -- 1. U-Duality Groups in Supergravity Theories -- 1.1. Noncompact exceptional groups as symmetries of maximally extended supergravity theories -- 1.2. Exceptional U-duality groups in matter coupled supergravity theories -- 2. U-duality Groups and Black Hole Entropy in Supergravity Theories -- 3. Conformal and Quasi-Conformal Realizations of Exceptional Groups -- 4. The Minimal Unitary Representations of Exceptional Groups -- Acknowledgments -- References -- Fusion Products of SLN Symmetric Power Representations and Kostka Polynomials R. Kedem -- 1. Introduction -- 2. The fusion product -- 3. The case of fundamental representations -- 4. The case of generic symmetric power representations -- Acknowledgments -- References -- Structure of Bi-Ideals Via Differential Calculi V. K . Kharchenko -- Acknowledgments -- References -- Virasoro-Type Algebras Associated with Higher-Rank Aperiodic Point Sets V. Mazorchuk and R. Twarock -- 1. Introduction -- 2. Aperiodic point sets via the projection method -- 2.1. A one-dimensional example -- 2.2. A two-dimensional example -- 3. Virasoro-type algebras -- 3.1. The one-dimensional case -- 3.2. The two-dimensional case -- 4. Concluding remarks -- Acknowledgments -- References -- Representations of the Poincar6 Group From Positive Energy Representations of S0(2,3) P. Moylan -- 1. Introduction -- 2. So0(2,3), the Poincar6 group and their Lie algebras -- 3. Algebraic results -- 4. Representations -- References -- III. Quantization and Quantum Gravity -- Theoretical Framework for Relativistic Resonances A. R. Bohm and Y. Sato -- References -- Nonlinear Schrodinger Equations for Particles and Antiparticles H.-D. Doebner and G. A. Goldin -- 1. INTRODUCTION.

2. A FAMILY OF NONLINEAR SCHRODINGER EQUATIONS DESCRIBING PARTICLES -- 3. STATIONARY SOLUTIONS -- 4. UPPER BOUND TO NONLINEARITY -- 5. PARTICLE VS. ANTIPARTICLE SYSTEMS -- Acknowledgments -- References -- Expanded Quantum Linear Harmonic Oscillator D. R. Finkelstein and M. Shiri-Garakani -- 1. Introduction -- 2. Time-independent oscillator -- 3. Clifford algebra representation -- Acknowledgments -- References -- Thermodynamic Limits, Non-Commutative Probability, and Quantum Entanglement J. F. Johnson -- 1. Introduction -- 2. The precise problem -- 3. Coherent superposition and Quantum entanglement -- 4. Where do we put the cut? -- 5. The physical content of the solution -- 6. Mathematical content -- References -- Canonical Reduction of Gravity: From General Covariance to Dirac Observables and Post-Minkowskian Background-Independent Gravitational Waves L. Lusanna -- References -- Where are the States of a Black Hole? S. D. Mathur -- 1. INTRODUCTION -- 2. AN ABSTRACT ARGUMENT ABOUT 'HAIR' -- 3. THE Dl-D5 SYSTEM -- 4. THE THREE CHARGE Dl-D5-P SYSTEM -- 5. FRACTIONATION -- 6. CONCLUSIONS -- ACKNOWLEDGMENTS -- REFERENCES -- What is Quantum Theory of Gravity? D. Minic and C.-H. Tze -- 1. What is Quantum Mechanics? -- 2. And What is Quantum Theory of Gravity? -- Acknowledgments -- References -- Quantizing Gravitational Collapse C. Vaz -- 1. Introduction -- 2. Quantization -- 3. The "eternal" black hole: Mass spectrum and entropy -- 4. Hawking radiation -- 5. Beyond the WKB approximation -- 6. Conclusions -- Acknowledgments -- References -- IV. D < 4 Field Theories -- String and Brane Worldvolumes -- Phase Structure of Non-Compact QED3 and the Abelian Higgs Model T. Appelquist and L. C. R. Wijewardhana -- 1. Introduction -- 2. QED3 and its Symmetry Breaking -- 3. A Conjectured Constraint -- 4. Lattice Studies -- 5. Gap Equation Estimates.

6 . Abelian Higgs Model -- 7. Summary and Discussion -- Acknowledgments -- References -- Plane Wave and Vortex-Like Solutions of Noncommutative Maxwell-Chern-Simons Theory D. Arnaudon, G. Alexanian and M. B. Paranjape -- 1. Introduction -- 2. Noncommutative Yand-Mills-Chern-Simons theory -- 3. Noncommutative plane waves -- 4. Static gauge solutions -- Acknowledgments -- References -- Quantum Hall Physics in String Theory 0. Bergman -- 1. Introduction -- 2. Branes and the 2d FQHE -- 2.1. The DO-brane quantum Hall fluid -- 2.2. The D.2-brane effective gauge theory -- 2.3. The quasihole/quasiparticle excitations -- 2.4. The matrix model -- 3. Branes and higher-dimensional FQHFs -- 3.1. Quasimembranes -- 4. Where do we go from here? -- Acknowledgments -- References -- Branes, Strings, and Odd Quantum Nambu Brackets T. L. Curtright and C. K. Zachos -- 1. Classical Nambu Dynamics -- 2. Quantum Nambu Dynamics -- Acknowledgments -- Appendix A. Solenoidal Flow -- References -- D Branes in 2d String Theory and Classical Limits S. R. Das -- 1. Overview -- 2. Preliminaries -- 3. Classical bosonization -- 4. Quantum bosonizat ion -- 4.1. Bosonization at the algebraic level -- 4.2. Nonrelativistic fermions in an inverted harmonic oscillator potential -- 5. Disconnected fermi fluids and quantum fluctuations -- 6. Explicit calculations -- Acknowledgments -- References -- Aspects of Diffeomorphism Invariant Theory of Extended Objects V , G. Gueorguieu -- Introduction. -- Homogeneous Lagrangians. -- First order homogeneous Lagrangians - canonical form. -- Extended objects. -- Summary. -- References -- Interaction Between Vortices in Models with Two Order Parameters R. MacKentie, M.-A. Vachon and U. F. Wichoski -- 1. Introduction -- 2. Review of previous work -- 3. Intervortex force -- Acknowledgments -- References.

Supersymmetric Boundary Conditions for the N = 2 Sigma Model I. V. Melnikov, M. R. Plesser and S. Rinke + -- 1. Introduction -- 2. Symmetries and Boundaries in Classical Field Theory -- 3. The NLSM with Boundaries -- 4. Conditions for = 2 Supersymmetry -- 5. Discussion -- Acknowledgments -- References -- Yang-Mills Theory, from 2 + 1 to 3 + 1 Dimensions V. P. Nair -- 1. Introduction -- 2. The 2 + 1 dimensional theory -- 3. The 3 + 1 dimensional theory -- Acknowledgments -- References -- Three Dimensional Simplicia1 Yang-Mills Theory: An Approach to the Mass Gap S. G. Rajeev -- 1. Hamiltonian Formulation of Yang-Mills Theory -- 2. Conformal and Symplectic Invariance -- 3. Reduced Yang-Mills Theory -- 4. Simplicial Sets and Complexes -- 5. Simplicial Gauge Theory -- 6. Quantization of Simplicial Gauge Theory -- 7. Subdivision of a Face -- Acknowledgments -- References -- V. D > 3 Field Theories and Gravity -- Cross-Order Relations in N = 4 Supersymmetric Gauge Theories C. Anastasiou, 2. Bern, L. J. Dixon and D. A. Kosower -- 1. Introduction -- Acknowledgments -- References -- BPS Domain Walls in Massive Hyper-Kahler Sigma Models M. Arai, M. Nitta and N. Sakai -- 1. Introduction -- 2. Kahler Sigma Models and Walls -- 3. Massive Hyper-Kahler Sigma Models and Walls -- 4. Massive Hyper-Kahler Models from n/ = 2 SUSY QCD -- 5. Domain Wall Solutions -- 6. Conclusions -- Acknowledgments -- References -- Higher Derivative Terms in N = 2 SUSY Effective Actions P. C. Argyres, A. Awad, G. A. Braun and F. P. Esposito -- 1. Introduction -- 2. Effective Actions -- 3. N = 2 Supersymmetry -- 4. Harmonic Superspace -- 5. Results -- 6. Superspace Chern-Simons-like Terms -- 7. Conclusions -- Acknowledgments -- References -- A Short Introduction to the Fate of the a-Vacuum H. Collins -- References.

Yangian Symmetry in D = 4 Superconformal Yang-Mills Theory L. Dolan, C. R. Nappi and E. Witten.