Howard Weiss - US grants

The Mathematical Sciences Postdoctoral Research Fellowships are awards to recent recipients of doctoral degrees. The fellowship is designed to provide 24 months of support divided into 18 academic months and 3 periods of two summer months. The recipient has the option of a Research Instructorship which allows the 18 months of academic support to be taken as 9 months of full-time support and 18 months of half-time support. Mathematical Sciences Postdoctoral Research Fellowship awards allow fellows to choose research environments that will have maximal impact on their future scientific development. Howard Weiss received his doctoral degree from the University of Maryland, and will pursue research in the area of pure mathematics, specifically, regularity of entropy for Anosov and Geodesic Flows and geometric rigidity, under the guidance of Anatole Katok at the California Institute of Technology.

9403724 Weiss The research covers several topics in smooth dynamical systems including hyperbolic sets in dynamical systems, relationships with thermodynamics, stochastic behavior of flows and related spectral theory. The research has potential applications to other areas of dynamical systems, chaos and areas of mathematical physics involving ergodicity and entropy. ***

Invariant sets of dynamical systems are not generally self-similar in the strict sense. However, in work with others, the PI has shown that in some important cases, these sets can be decomposed into subsets each possessing a type of scaling symmetry. Sets which admit such structure are called multifractals (MF). This proposal involves the continuing investigation of the fine structure of these multifractals and an attempt to use the MF analysis to give new insights into dynamical systems and possibly yield a new (physical) classification of dynamical systems. In addition, the proposed work involves various problems in dimension theory which arise in mathematical biology as well as research on the relations between non-negative curvature and complicated dynamics of the geodesic flow. Regarding the latter, the (in)famous (xy)^2 Hamiltonian system, a model for a classical Yang-Mills field, which is orbit equivalent to a geodesic flow on a non-negative curved surface will be studied. Many physical and biological systems (including turbulent fluids, root systems of plants, stressed pieces of metal, NMR images of the brain, clouds, and galaxies in the universe) seem to possess some type of complicated fractal structure. Mathematically such objects are called multifractals. In previous work, the principal investigator presented a rigorous mathematical foundation for the study of some important classes of multifractals. The plan is to extend this work to larger classes of systems and to use this mathematical analysis to help understand the underlying physical or biological systems. The PI is particularly interested in applications to plant biology. In a different area, a large class of physical systems which are central to celestial mechanics and plasma physics can be studied by first transforming them to a ''geometric system'' called a geodesic flow and then studying the geodesic flow. In previous work, the PI showed that a large class of these flows, which some thought were easily understood and mathematically and physically boring, have extremely complicated behavior and are in fact chaotic. The investigations into some specific examples including an example from gauge field dynamics, which is one of the central theoretical problems in particle physics will be continued.

The proposed project has several related components: 1) We
plan to continue our study of the Schelling segregation
model as a dynamical system. This model, which first arose
in economics, is related to a number of lattice models in
statistical physics like the lattice gas, but more difficult
due to the inherent non-local nature of site coupling; 2) We
plan to study the "rigidity" of periodic point invariants
for symbolic and hyperbolic dynamical systems. These
topological invariants include, for a Holder continuous
function f, the unmarked periodic orbit spectrum, the beta
function P(-s f), and the zeta function. These invariants
are fundamental objects of study in dynamics and statistical
physics, but the information about the function f they
capture is subtle and poorly understood; 3) We plan to
continue our investigation into the distribution of values
of fundamental quantities in ergodic theory (e.g. Lyapunov
exponents, local entropy, and Birkhoff averages) and the
fine structure of the corresponding phase space
decomposition.

The proposed project has several related components: 1) We
plan to continue our study of the Schelling segregation
model as a dynamical system. This model, which was first
proposed by the eminent economist Thomas Schelling, is
related to a number of lattice models in statistical physics
like the lattice gas, but more difficult due to the inherent
non-local nature of site coupling; 2) Pressure is a
fundamental object of study in statistical physics, but even
in highly idealized systems, the information about the
system it captures is subtle and poorly understood. We plan
to study whether certain systems are completely identified
by their pressure. These problems have striking similarities
to fascinating questions which Kac adroitly summarized with
the question "Can you hear the shape of a drum?"; (3) For
ergodic systems, the time average of a function along almost
every orbit equals the spatial average. Only very rarely can
almost every orbit be replaced by every orbit. We plan to
study the fine structure and dimension of the exceptional
set whose time average does not coincide with the spatial
average

0104675
Levi

This U.S. Mexico award will support Drs. Mark Levi, Howard Weiss, and Yakov Pesin in a research collaboration with Valentine Afraimovich of the Mathematics Department of the Unversidad Autonoma de San Luis de Potosi. The investigators plan to study: 1) Coupled map lattices corresponding to partial differential equations from physics and biology, in particular, the FitzHugh-Nagumo equation (which is of great interest in neurobiology). The collaborators intend to describe the ergodic properties of its local map and construct SRB measures for the attractor of this map. 2) The transition from coupled map lattices to partial differential equations via traveling waves, which will build a foundation for numerical modeling of some partial differential equations of evolution type. 3) The Dynamics of chains of coupled oscillators, in particular, those associated with the Sine-Gordon equation.

A coupled map lattice is a discrete time dynamical system whose phase space is of a particular form, and for which the overall system exhibits translational symmetry.
Coupled map lattices have recently gained wide popularity as models of spatio-temporal chaos and coherent structures. There is currently great interest in using coupled map lattices to model turbulence, nerve cells, phase transitions in statistical physics, and crystals. They also arise naturally from the discrete version of evolution partial differential equations.

ABSTRACT
Weiss

This proposal is to study several applications of dynamical systems to
statistical physics, geometry, and population biology/ecology. (i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, natural classes of Holder continuous potentials for certain one-dimensional lattice spin systems are determined by their free energy. We also plan to investigate striking similarities between the rigidity of free energy and fascinating rigidity problems in spectral geometry and number theory. (ii) Little is known about the
dynamics of the geodesic flow on positively curved manifolds. The PI
plans to continue studying the relations between positive curvature and
complicated dynamics of the geodesic flow. (iii) The PI has started a
program to systematically study the global dynamics and bifurcations for
nonlinear Leslie models where the fertility rates and survival
probabilities have various natural functional forms as functions of the
population size.

(i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. Lattice spin systems provide an important and illuminating family of models in statistical physics, condensed matter physics, and chemistry. For instance, phase transitions correspond to non-differentiability for some derivative of free energy. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, potentials for
one-dimensional lattice systems are determined by their free energy. We
hope this work will provide new insights into this important, yet
mysterious, quantity. (ii) Essentially all demographic and animal
population models in current use are based on the linear Leslie model.
Many population biologists, ecologists, and demographers are now looking
to nonlinear population models for more accurate population forecasting.
The PI has started a program to systematically study the global dynamics
and bifurcations for nonlinear Leslie models where the fertility rates and
survival probabilities have various natural functional forms as functions
of the population size. One of our ultimate goals is to create a
``population modeling toolbox'' which could be used by a wide range of
population modelers to more accurately predict animal populations.