Mathematical Physics Seminar
Rutgers University
Hill Center, Room 705
October Schedule

Organized by: Joel L. Lebowitz
lebowitz@math.rutgers.edu

Please join us every Thursday at 11:45am in the kitchen of room 705 for coffee and cookies

Speaker: G. Gallavotti, INFN-Italy
Date: 10/5/06
Time & Place: 12-1pm ; Hill 705
Title: "Invariant tori in Hamiltonian mechanics: a review"
Abstract: Invariant tori in systems close to anisochronous integrability exist with any dimension. The maximal ones are the KAM tori and have good analiticity properties. This does not seem to be the case for lower dimensional tori although a perturbation approach to their study is possible. Analiticity properties can be established but, so far, are weaker than convergence of the perturbation series: sometimes Borel-summability can be proved but often the analiticity that can be proved is even weaker.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM

Speaker: M. Stenlund, University of Helsinki
Date: 10/5/06
Time & Place: 2-3pm ; Hill 705
Title: "Pendulum, Homoclinic Splitting and Exponential Smallness"
Abstract: I will consider a Hamiltonian system describing a pendulum weakly coupled with an arbitrary number of anharmonic oscillators. A large proportion of the phase space is filled by invariant tori supporting regular motion with bounded trajectories and action variables close to those in the uncoupled case.
It is well known that transversal intersections of the unstable manifolds with stable manifolds of such tori not only induce chaos but can lead to global instability through drift of actions. One is lead to study the intersection "angle" between the manifolds. I will explain why the latter is (at most) exponentially small with respect to the forcing rapidity.

PLEASE NOTE THERE WILL BE NO SEMINAR THURSDAY OCTOBER 12, 2006

Please join us every Thursday at 11:45am in the kitchen of room 705 for coffee and cookies

Speaker: K. Mischaikow, Rutgers University
Date: 10/19/06
Time & Place: 12-1pm ; Hill 705
Title: "Computational Homology and Spatially Explicit Nonlinear Systems"
Abstract: Patterns produced by nonlinear systems and observed either through experiment or numerical simulation can be thought of as projections from an infinite dimensional phase space onto a finite (even discrete) observation space. However, since this observation space is still typically very high dimensional, it is often necessary to further simplify the data. In this talk I will argue that even the most brutal of reductions - the homology of the patterns - still allows one to recover interesting information about the dynamics of the original system. Furthermore, because of the discrete nature of homology, this reduction raises interesting open mathematical questions related to the interpretation of the reduced data.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM

Speaker: A. Toom, Federal University of Pernambuco
Date: 10/19/06
Time & Place: 2-3pm ; Hill 705
Title: "Every Continuous Operator Has An Invariant Measure"
Abstract: We prove existence of invariant measure for a large class of random processes with discrete time without assuming their linearity. Our main examples are "processes with variable length", in which components may appear and disappear in the course of functioning. One of these examples displays a version of 1-D non-ergocidity.

Speaker: M. Bramson, University of Minnesota
Date: 10/26/06
Time & Place: 12-1pm ; Hill 705
Title: "Exclusion Processes in One and Higher Dimensions"
Abstract: Exclusion processes form one of the major classes of interacting particle systems. There, particles on a lattice execute independent random walks in continuous time, except when the target site is already occupied, in which case the particle remains at the original site.
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Many results exist for the lattice Z. In particular, the equilibria of such exclusion processes are in many cases well understood. Little is presently known, however, for Z^d, d>1 . We will review the behavior of the exclusion process on Z and with the remaining time present the foundation of a theory for d>1 .

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM

Speaker: Y. Sinai, University of Princeton
Coauthor: D. Li, University of Princeton
Date: 10/26/06
Time & Place: 2-3pm ; Hill 705
Title: "Blow ups of complex solutions of 3-dim Navier-Stokes system and the Renormalization Group Method"
Abstract: We consider complex-valued solutions of the 3-dim Navier-Stokes System without external forcing on R3.We show that there exists an open set in the space of 10-parameter families of initial conditions such that for each family from this set there are values of parameters for which the solution develops blowup in finite time.