Speaker: H, Brezis, Rutgers University
Time/Place:Thursday 11/4/05, 12:00-1pm
Title: "Reduced Measure"
Abstract: Consider a nonlinear elliptic PDE of the form $$ -\Delta u + g(u) = \mu\tag 1 $$ where $g$ is an increasing function and $\mu$ is a given measure. If the measure $\mu$ is too ``concentrated'' equation (1) need not have a solution. However we will prove that there is a natural concept of generalized solution which is stable under all natural approximations. In some sense, ``this is the best one can do'' in the absence of a solution.

Speaker: N. Zhangi, Rutgers University
Time/Place:Thursday 11/4/05, 2-3PM
Title: "On the Distribution of the Wave Function for Systems in Thermal Equilibrium"
Abstract: A density matrix that is not pure can arise, via averaging, from many different distributions of the wave function. This raises the question, which distribution of the wave function, if any, should be regarded as corresponding to systems in thermal equilibrium as represented, for example, by the density matrix ?= (1/Z) exp (-H) of the canonical ensemble. To answer this question, we construct, for any given density matrix ?, a measure on the unit sphere in Hilbert space, denoted GAP (?), using the Gaussian measure on Hilbert space with covariance ?. We argue that GAP (?) corresponds to the canonical ensemble. Key words: canonical ensemble in quantum theory; probability measures on Hilbert space; Gaussian measures; density matrices.

Speaker:A. Guiliani, Princeton University
Time/Place:Thursday 11/10/05, 12-1PM
Title: "The weak coupling 2D Hubbard model at exponentially small temperatures"
Abstract:The Hubbard model is the simplest possible model in the study of correlated electrons. Still, only a few rigorous results are known on its behavior at zero or non zero temperature. It is believed that the system shows a (superconducting?) instability for temperatures smaller than an exponentially small one and that it behaves like a Fermi or like a non Fermi liquid for larger temperatures, depending on the choice of the density. In this talk I will discuss how the free energy and the two point correlation function of the weak coupling underdoped 2D Hubbard model can be computed for temperatures larger than an exponentially small one in terms of a (renormalized) convergent expansion. The construction shows that the wave function renormalization is approximately temperature independent in the considered range of temperatures, and this can be interpreted by saying that the system is a Fermi liquid in the considered range of temperatures.

Speaker:A. Soffer, Rutgers University
Time/Place:Thursday 11/10/05, 2-3:00PM
Title: "Long range nonlinear scattering"
Abstract: when a dispersive or hyperbolic wave equation is perturbed nonlinearly the asymptotic solutions may change character. In particular, if the nonlinear term is slowly vanishing near zero amplitude, the asymptotic behavior is going to be "modified free".

I shall present a new approach to deal with this problem for both NLS and NLKG equations.

Speaker:E. Speer, Rutgers University
Time/Place:Thursday 11/17/05, 12-1:00PM
Title: "Entropy of an Open Lattice System"
Abstract:We discuss the behavior of the Gibbs-Shannon entropy of the stationary nonequilibrium measure describing a one-dimensional lattice gas governed by symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit the leading order behavior of this entropy has been shown by Bahadoran to be that of a product measure corresponding to strict local equilibrium. We compute the first correction. This entropy correction depends only on the scaled truncated pair correlation, which describes the covariance of the density field, and coincides, in the hydrodynamic limit, with the corresponding correction obtained from a Gaussian measure with the same covariance.

Speaker:M. Kiessling, Rutgers University
Time/Place:Thursday 11/17/05, 2-3:00PM
Title: "Microscopic derivation of a scalar caricature of the relativistic Vlasov-Maxwell equations"
Abstract:I present results obtained jointly with Yves Elskens (Marseille) and Valeria Ricci (Palermo). We prove a law of large numbers for the Vlasov equations obtained by writing the Vlasov-Maxwell equations using the electromagnetic potentials $\phi$ and $A$ and then discarding all terms involving $A$, or not quite: the electromagnetic coupling between matter and fields is regularized to avoid UV divergence problems. Interestingly, the resulting system is physically rather a wave-type generalization of a Newtonian gravitating system than a plasma. As a byproduct of our proof we obtain existence and uniqueness of measure solutions in a suitable Banach space topology. If time permits, I also talk about what we know about the fluctuations around the mean.