This volume continues the series of proceedings of summer schools on theoretical physics related to various aspects of the structure of condensed matter, and to appropriate mathematical methods for an adequate description. Three main topics are covered: (i) symmetric and unitary groups versus electron correlations in multicentre systems; (ii) conformal symmetries, the Bethe ansatz and quantum groups; (iii) paradoxes of statistics, space-time, and time quantum mechanics. Problems considered in previous schools are merged with some new developments, like statistics with continuous Young diagrams, the existence and structure of energy bands in solids with fullerenes, membranes and some coverings of graphite sheets, or vortex condensates with quantum counterparts of Maxwell lows.
Part 1 Symmetric and unitary groups vs. electron correlations in multicentre systems: at the start of a new golden age of physics, B.G. Wybourne; Weyl's denominator identity and its deformations, R.C. King; representations of the Dirac algebra for a constrained system, Y. Ohnuki; meta-symmetry, Y.I. Granovskii. Part 2 Conformal symmetries, Bethe-Ansatz and quantum groups: quantum phenomena with vortex condensates, A. Vourdas; statistical and group properties of the fractional quantum Hall effect, B.G. Wybourne; braids for pretzel knots, M. Suffczy-ski; does the Bethe-Ansatz result in a complete set of stationary states for Heisenberg rings?, W.J. Caspers et al; string configuration on small rings, D. Goojuch et al. Part 3 Paradoxes of statistic, space-time and quantum mechanics: elementary energy bands in crystalline solids - space groups with 3-dimensional strata only, L. Michel and J. Zak; invariant theory in crystal symmetry, J.S. Kim et al; crystal symmetry and time scales, V.I. Yukalov and E.P. Yukalov; elastic instabilities of cubic media, T. Paszkiewicz et al; wavelet multiresolution for the Fibonacci chain, M. Andrle. (Part contents).