Abstract:
The phase diagram of the unconstrained $t-J$ model is calculated using the random phase approximation. It is found that the extended $s$ and the $d_{x^2-y^2}$-channels are {\em not} degenerate near half filling. Extended $s$-pairing with a low $T_c$ occurs only for a band containing less then 0.4 electrons or holes per unit cell, whereas in a large region around half-filling $d$-wave pairing is the only stable superconducting solution. At half filling superconductivity is suppressed due to the formation of the anti-ferromagnetic Mott-Hubbard insulating state. By extending the analysis to the unconstrained $t-t'-J$ model, it is proven that, if a Fermi surface is assumed similar to the one that is known to exist in cuprous oxide superconductors, the highest superconducting $T_c$ is reached for about $0.7$ electron per site, whereas the anti-ferromagnetic solution still occurs for $1$ electron per site. It is shown, that the maximum $d$-wave superconducting mean field transition temperature is half the maximum value that the Ne\`el temperature can have in the Mott-insulating state.

Abstract:
The optical conductivity along and perpendicular to the planes is calculated assuming strong k-dependence of the scattering rate and the c-axis hopping parameter. Closed analytical expressions for the optical conductivy along these directions are obtained, and are shown to be integrable at low and high frequencies. A large and qualitatively different frequency dependence for both polarizations follows directly from the model. The expression for the in-plane conductivity has an effective scattering rate proportional to frequency, and can be easily generalized to provide a simple analytical expression, which may replace the Drude formula in the case of non-Fermi liquids.

Abstract:
The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding band, where the electrons experience besides the long-range Coulomb repulsion an on-site Hubbard U repulsion and a nearest-neighbour attractive interaction. {}From numerical calculations we see, that the collective mode spectrum now becomes particularly rich. Several branches can occur below the continuum of quasi-particle excitations, corresponding to order-parameter fluctuations of various symmetries of pairing, and collective spin-density fluctuations. {}From the collective mode softening near the nesting vectors it is concluded, that in the d-wave paired state an instability occurs toward the formation of a spin-density wave.

Abstract:
An introduction is given to collective modes in layered, high Tc superconductors. An experimental demonstration is treated of the mechanism proposed by Anderson whereby photons travelling inside the superconductor become massive, when the U(1) gauge symmetry is broken in the superconductor to which the photons are coupled. Using the Ferrell-Tinkham sumrule the photon mass is shown to have a simple relation to the spectral weight of the condensate. Various forms of Josephson plasmons can exist in single-layer, and bi-layecuprates. In the bi-layer cuprates a transverse optical plasma mode can be observed as a peak in the c-axis optical conductivity. This mode appears as a consequence of the existence of two different intrinsic Josephson couplings between the CuO2 layers. It is strongly related to a collective oscillation corresponding to small fluctuations of the relative phases of the two condensates, which has been predicted in 1966 by A.J. Leggett for superconductors with two bands of charge carriers. A description is given of optical data of the high Tc cuprates demonstrating the presence of these and similar collective modes.

Abstract:
The f-sum rule is introduced and its applications to electronic and vibrational modes are discussed. A related integral over the intra-band part of sigma(omega) which is also valid for correlated electrons, becomes just the kinetic energy if the only hopping os between nearest-neighbor sites. A summary is given of additional sum-rule expressions for the optical conductivity and the dielectric function, including expressions for the first and second moment of the optical conductivity, and a relation between the Coulomb correlation energy and the energy loss function. It is shown from various examples, that the optical spectra of high Tc materials along the c-axis and in the ab-plane direction can be used to study the kinetic energy change due to the appearance of superconductivity. The results show, that the pairing mechanism is highly unconventional, and mostly associated with a lowering of kinetic energy parallel to the planes when pairs are formed. Keywords: Optical conductivity, spectral weight, sum rules, reflectivity, dielectric function, inelastic scattering, energy loss function, inelastic electron scattering, Josephson plasmon, multi-layers, interlayer tunneling, transverse optical plasmon, specific heat, pair-correlation, kinetic energy, correlation energy, internal energy.

Abstract:
A many-body wavefuction is postulated, which is sufficiently general to describe superconducting pair-correlations, and/or spin-correlations, which can occur either as long-range order or as finite-range correlations. The proposed wave-function appears to summarize some of the more relevant aspects of the rich phase-diagram of the high-Tc cuprates. Some of the states represented by this wavefunction are reviewed: For superconductivity in the background of robust anti-ferromagnetism, the Cooper-pairs are shown to be a superposition of spinquantum numbers S=0 and S=1. If the anti-ferromagnetism is weak, a continuous super-symmetric rotation is identified connecting s-wave superconductivity to anti-ferromagnetism.

Abstract:
Classically the interaction between light and matter is given by the Maxwell relations. These are briefly reviewed and will be used as a basis to discuss several techniques that are used in optical spectroscopy. We then discuss the quantum mechanical description of the optical conductivity based on the Kubo formalism. This is used as a basis to understand how strong correlation effects can be observed using optical techniques. We will discuss the use of sum rules in the interpretation of optical experiments. Finally, we describe the effect of including interactions between electronic and collective degrees of freedom on optical spectra.

Abstract:
We discuss the possible existance of transverse optical plasma modes in superlattices consisting of Josephson coupled superconducting layers. These modes appear as resonances in the current-current correlation function, as opposed to the usual plasmons which are poles in the density-density channel. We consider both bilayer superlattices, and single layer lattices with a spread of interlayer Josephson couplings. We show that our model is in quantitative agreement with the recent experimental observation by a number of groups of a peak at the Josephson plasma frequency in the optical conductivity of La$_{1.85}$Sr$_{0.15}$CuO$_4$

Abstract:
We present preliminary results of an HST/NICMOS program to image merger remnants in the J, H and K bands. The nuclear brightness profiles for most sample galaxies are similar to those typical for elliptical galaxies, but some (including the well-studied NGC 3921 and 7252) have an unusually high luminosity density at small radii. This is consistent with the prediction of N-body simulations that gas flows to the center during a merger and forms new stars.

Abstract:
A detailed calculation is presented of the dielectric function in superconducttors consisting of two Josephson coupled superconducting layers per unit cell, taking into account the effect of finite compressibility of the electron fluid. From the model it follows, that two longitudinal, and one transverse optical Josephson plasma resonance exist in these materials, for electric field polarization perpendicular to the planes. The latter mode appears as a resonance in the transverse dielectric function, and it couples directly to the electrical field vector of infrared radiation. A shift of all plasma frequencies, and a reduction of the intensity of the transverse optical Josephson plasmon is shown to result from the finite compressibility of the electron fluid.