Mathematical Physics Seminar
September Schedule
Organizer- Joel L. Lebowitz
email: lebowitz@math.rutgers.edu
- Speaker- C. Woodward, Rutgers University
- Title-
Sympletic Geometry, Non-Euclidean Tetrahedra, and Racah-Wigner Symbols
- Time/place- 9/12/2002 11:30am in Hill 705
- Abstract- 6j symbols were invented by Racah as a way of simplifying
atomic spectroscopy calculations for many-electron systems. Wigner and
Ponzano-Regge proposed beautiful formulas for the classical limit of these
symbols, involving Euclidean tetrahedra. One of these formulas was
recently proved by J. Roberts. I will talk about a new proof of this
formula, using functorial properties of Bohr-Sommerfeld sequences,
extensions to non-Euclidean tetrahedra/"quantum" groups, and relation to
classical limits of "quantum" three-manifold invariants.
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- D. Zeilberger, Rutgers University
Title- Fancy Toy Models for Self-Avoiding Walks
Time/place- 9/12/2002 1:30pm in Hill 705
Abstract-
The Counting of Self-Avoiding Walks is probably intractable,
so we have to compromise and study toy model. I will describe one such,
somewhat fancy toy model, using the Umbral Transfer Matrix Method.
Speaker- H. Brezis, University of Paris/Rutgers University
Title- New Results from the Ginzburg-Landau Model in 3-D
Time/place- 9/19/2002 11:30am in Hill 705
Abstract-I will report on a recent joint work with
J. Bourgain and P. Mironescu.
Consider a domain $G$ in $\Bbb R^3$ and a boundary data $g$ in the
Sobolev space $H^{1/2}(\partial G; S^1)$. To every such $g$ we attach
a distribution $T(g)$ of the form $\sum_i(\delta_{P_i} -
\delta_{N_i}), P_i, N_i \in \partial G$ and a number $L(g); T(g)$
describes the location of the singularities of $g$ and $L(g)$ is the
length of a minimal connection connecting the singularities. Such
objects play a fundamental role in the study of the Ginzburg-Landau
energy
E_\varepsilon (u) = \frac{1}{2} \int\limits_G |\nabla u|^2 +
\frac{1}{4\varepsilon^2} \int\limits_G (|u|^2 - 1)^2
as $\varepsilon \to 0$, where $u$: $G \to \Bbb C$ and $u=g$ on
$\partial G$. For example, the minimum energy is of the order of $\pi
L(g)\log (1/\varepsilon)$ and minimizers concentrate along vortex
lines in $G$ emanating from $P_i$'s and terminating at the $N_i$'s.
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- R. Nussbaum, Rutgers University
Title- Differential-Delay Equations: Some Unexpected Connections
Time/place- 9/19/2002 1:30pm in Hill 705
Abstract-We shall discuss joint work with John Mallet-Paret
in which we consider nonlinear differential equation
$$
ax^\prime(t) = f(x(t),x(t-r)), \quad r = r(x(t)) \eqno(1)
$$
Here $f$ and $r$ are given functions and $a > 0$. A simple looking example
to which our theory applies is
$$
ax^\prime(t) = -x(t) - k x(t-r), \quad r = 1 + c x(t), \eqno(2)
$$
where $a > 0$, $c > 0$, and $k > 1$. Under appropriate assumptions on $f$
and $r$ we know that for all sufficiently small $a > 0$ (1) has a
``slowly-oscillating periodic solution'' (which depends on $a$). We are
interested in the limiting shape of the graphs of such solutions as $a$
approaches zero. In answering this question we are led to the study of the
so-called max-plus equation
$$
x(t) + p = max\{k(s,t) + x(t): c(t) \leq s \leq d(t), \quad 0 \leq t \leq
L\}.\eqno(3)
$$
In (3), $k$, $c$ and $d$ are given continuous functions with $0 \leq c(t)
\leq d(t) \leq L$ for $0 \leq t \leq L$, and one seeks an ``additive
eigenvector $x(t)$'' and ``additive eigenvalue $p$'' which solve eq. (3).
The presence of function $c$ and $d$ makes eq. (3) much more subtle than in
the case $c(t):=0$ and $d(t):=1$.
Speaker- H. Furstenberg, Hebrew University
Title- Growth Rate of Trees, Eigenmeasures, and
Transversality of Fractals
Time/place- 9/26/2002 11:30am in Hill 705
Abstract- Two fractals, A and B, in Euclidean space will be
called transversal if any translate of one meets the other in a set of
Hausdorff codimension at least equal to the sum of the codimensions of
A and B. We will show how this question for certain pairs A,B relates
to the growth rate of trees, and this in turn to a certain eigenvalue
equation for operators on measures. We will discuss implications for
sizes of orbit closures in dynamical systems.
Please note there will be a brown bag lunch between the 2 seminars
this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- J. Stark, University College London
Title- Inertial Manifolds and Moment Approximations for
Stochastic Processes
Time/place- 9/26/2002 1:30pm in Hill 705
Abstract-It has been observed that for some stochastic
processes
good
approximations can be obtained by assuming that the distribution is
of a particular form, eg normal, negative binomial etc. A good
example occurs in recent models of host-parasite interaction
developed by Isham and her collaborators. The assumption of a
particular form of the distribution can be interpreted as a
functional relation amongst the moments and defines a manifold in the
space of all distributions. If the approximation is a good one, one
would expect this manifold to be approximately invariant and to be
attracting. This is very similar to the concept of an inertial
manifold which in the last decade has had a profound influence on the
study of a number of important partial differential equations (such
as the Navier-Stokes and Ginzburg-Landau equations, and some reaction
diffusion equations). This talk will give an introduction to inertial
manifolds and then go on to explore the relationship of this concept
to moment approximations in a particular host-parasite model.