Atoms in a standing wave of light experience a periodic potential called an optical lattice. The study of cold atoms in an optical lattice has become a major frontier of cold atom physics over the last few years. We propose to open a new domain of this study by having the standing waves within an optical cavity. This is expected to induce new collective effects through the common coupling of the atoms to a single photon mode. For example, we anticipate that collective effects within a cavity can be expected to cool a large number N of atoms or molecules to very low temperatures. As our initial theoretical [1,2] and first incomplete experimental studies [3,4,5] show, maximum cooling rates are expected in the presence of a red-detuned laser field and when the cavity leakage rate is as large as the square root of N times the single-particle coupling constants. The phonons, describing the movement of the particles inside the optical lattice potential, are then continuously converted into photons, which leak out through the cavity mirrors. The result is an evaporation of the kinetic energy in the system on a time scale given by the very large leakage rate of photons through the resonator mirror. We propose to study the cooling and related collective effects that are expected to occur naturally in these nonlinear quantum systems and to compare experimental results with detailed and newly developed theoretical models. The aim of the proposal is to begin a study of these effects through a collaboration of experimenters and theorists, bringing together expertise in cold atoms, cavity QED, quantum physics, many body systems and non-linear dynamics.  A. Beige, P.L. Knight, and G. Vitiello, Cooling Many Particles at Once, New J. Phys. 7, 96 (2005). A. Beige, P.L. Knight, and G. Vitiello, Cooling many particles to very low temperatures, Braz. J. Phys. 35, 403 (2005). J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, Quantum Nondemolition Measurements using Cold Trapped Atoms, Phys. Rev. Lett. 78, 634 (1997).  H. W. Chan, A. T. Black, and V. Vuletic, Observation of Collective-Emission-Induced Cooling of Atoms in an Optical Cavity, Phys. Rev. Lett. 90, 063003 (2003). Private communication with Ph. Grangier.