Speaker: Cedric Villani, ENS, France Date/Time/Place: Thursday, Jan. 29, 2009, 12:00pm, Hill 705 Title: Mathematical Theory of Landau Damping
Abstract: The Landau damping is a famous and mysterious
phenomenon in plasma physics and galactic dynamics. Although well
understood at the linear level, it has remained essentially untouched
in the physically relevant nonlinear setting. I shall describe a
current work with Clement Mouhot which is aimed at solving this
problem.
Speaker: H. Brezis, Rutgers University
Date/Time/Place: Thursday, Feb. 5, 2009, 12:00pm, Hill 705 Title: On a Conjecture of J. Serrin
Abstract: In 1964 J. Serrin proposed the following conjecture.
Let u be a weak solution (in W^{1,1}) of a second order elliptic
equation in divergence form, with Holder continuous coefficients, then
u is a classical solution. We announce a solution of this conjecture
assuming even weaker conditions on u and on the coefficients.
THERE WILL BE A BROWN BAG LUNCH ON FEB. 5TH FROM 1:00-2:00PM
BETWEEN SEMINARS - PLEASE JOIN US.
Speaker: Michael Aizenman, Princeton University
Date/Time/Place: Thursday, Feb. 5, 2009, 2:00pm, Hill 705 Title: Localization Bounds for Multiparticle Systems
Abstract: The talk will focus on the extension of the
mathematical theory of Anderson localization to quantum systems of N
interacting particles on lattices of arbitrary dimension. The main
result is that for all N there are regimes of high disorder, and/or
weak enough interactions, for which the system exhibits spectral and
dynamical localization. The localization is expressed in terms of
bounds on the transition amplitudes, and more generally eigenfunction
correlators, which decay at exponential rate in terms of the Hausdorff
distance in the configuration space, uniformly in the allotted
time. The bounds are derived through the analysis of fractional
moments of the N-particle Green function. (Joint work with Simone Warzel).
Speaker: Irene Gamba, University of Texas at Austin Date/Time/Place: Thursday, Feb. 12, 2009, 12:00pm, Hill 705 Title: Kinetic Evolution for Complex Multi-linear Particle Interactions Abstract: We present a generalized formulation of kinetic non-local collisional models of Maxwell type that cover a large class of global energy dissipative phenomena, such as inelastic collisions, mixtures and slowdown cooling processes, economics and social dynamics; in the setting of multiplicatively interactive stochastic processes. The working framework is in the space of characteristic functions of probabilities measures and is done for a class of equations of non-local multi linear form with basic symmetries and invariance, where is possible to recover the longtime dynamics as those of metrics evolution to stable laws characterized by self-similar states for finite or infinity energy initial data. These are states in probability space which can not have all moments bounded and even may admit singularities at the origin, while remaining integrable, as in a simple example of limit mixture model for a slow-down process.
Speaker: Pavel Dubovskiy, Stevens Institute of Technology, N.J. Date/Time/Place: Thursday, Feb. 19, 2009, 12:00pm, Hill 705 Title: Solvability of the spatially inhomogeneous
coagulation-fragmentation equation Abstract: Existence and uniqueness of a continuous global
solution is proved for the spatially inhomogeneous Smoluchowski
coagulation equation with fragmentation taken into account. The
coagulation kernels may be unbounded. The proof is based on new a priori estimates.
Speaker: Emil Yuzbashyan, Rutgers University Date/Time/Place: Thursday, Feb. 26, 2009, 12:00pm, Hill 705 Title: The link between integrability, level crossings, and
exact solution in quantum models Abstract: I will discuss
the connection between energy level crossings in integrable systems
and their integrability. In particular, I will consider a general
quantum Hamiltonian linear in the coupling u, H(u) = T + uV, and
require that it has the maximum possible number of nontrivial
commuting partners similarly linear in u. It turns out that this
commutation requirement alone leads to: (1) an exact solution for the
energy spectrum and (2) level crossings, which are always present in
such Hamiltonians in violation of the famous Wigner-von Neumann
non-crossing rule. Moreover, these Hamiltonians can be constructed
explicitly by resolving the above commutation requirement and turn out
to be equivalent to so-called Gaudin magnets. In contrast, fewer than
the maximum number of conservation laws does not guarantee level
crossings.