Statistical Graph Theory

Award Number: 100784

Program(s): ALGEBRA,NUMBER THEORY,AND COM

Principal Investigator: Komlos, Janos

Co-PI Name(s): Endre Szemeredi

PI Email Address: komlos@math.rutgers.edu

Abstract: Graph Theory is a central subject of Discrete Mathematics, important both for theoretical reasons and in view of its wide applicability. It models patterns (and strength) of connections between the components of complex systems. Investigation of very large graphs usually require special mathematical tools.

The most widely known example for very large graphs is the connection structure of the Internet, where various parts differ substantially. Breaking up such large graphs to a small number of more homogeneous and thus more manageable pieces is the core of the PI's current research.

Spectroscopy and Electromagnetic Moments of Nuclei under Extreme Conditions

Award Number: 98800

Program(s): NUCLEAR THEORY, HEAVY ION NUCLEAR SCIENCE, INTERMEDIATE ENERGY NUCLEAR SC

Principal Investigator: Koller, Noemie

Co-PI Name(s): Jolie Cizewski

PI Email Address: nkoller@physics.rutgers.edu

Abstract: This proposal will investigate the structure of atomic nuclei at the extremes of stability, angular momentum, and energy. The studies will include nuclear systems beyond the proton drip line and at the

limits of stability to fission. Techniques will be developed to study light-ion transfer reactions on nuclei far from stability using radioactive ion beams. This proposal also addresses, via measurements of magnetic moments of excited states, the the microscopic description of nuclear wavefunctions as a function of energy and spin, the study of residual nucleon-nucleon interactions, and the interplay of single particle configurations with an underlying spherical or deformed core. A new technique, combining the traditional transient field method with the newly developed approach involving Coulomb excitation of {\it beams} in inverse kinematics, will be explored in order to measure magnetic moments of excited states with

subpicosecond lifetimes in nuclei close to shell closure. The structure of the proposed activities is designed to have great impact on the education and training of graduate and undergraduate students, as well as postdoctoral associates involved with these experimental research activities.

Kinematics of Stellar Systems: Distances, Mass Distributions, and Binary Star Frequencies

Award Number: 98650

Program(s): STELLAR ASTRONOMY and ASTROPHYSC

Principal Investigator: Pryor, Carlton

Co-PI Name(s): Theodore Williams

PI Email Address: pryor@physics.rutgers.edu

Abstract: "Kinematics of Stellar Systems: Distances, Mass Distributions and Binary Star Frequencies"

AST-0098650

This project investigates the structure, stellar content and dynamical evolution of globular clusters and of the dark matter content of the dwarf spheroidal companion galaxies of our own Milky Way. Advantage is taken of the enormous amount of data that can be obtained with the Rutgers Fabry-Perot imager. This instrument is one of the few that can acquire the wide-field radial velocity data which are required to address these challenges. Combining Fabry-Perot measurements of the radial velocity distribution with HST determinations of the distribution of proper motions for member stars will yield direct geometrically based measures of the distances to 13 globular clusters. These new distances will have uncertainties about one-fifth as large as the currently accepted values. Repeated velocity measurements are one of the most productive ways of discovering binary stars and the greatly increased numbers which are expected will test models for the dynamical evolution of globular cluster binaries. The project will use three-color stellar photometry with the wide-field imagers to search for tidal tails of stellar debris associated with the Draco dwarf spheroidal galaxy. Radial velocity measurements of the stars thus found can yield total dark matter masses for the dwarf spheroidal galaxy and the rate at which stars are being lost from the system. Thus, the work done under this award will impact upon several areas of astrophysics including the extragalactic distance scale and the characteristics of dark matter in dwarf spheroidal galaxies.

Interactions of Logic with Group Theory and Combinatorics

Award Number: 100794

Program(s): FOUNDATIONSINFRASTRUCTURE PROGRAM

Start Date: 6/1/2001

Principal Investigator: Cherlin, Gregory

Co-PI Name(s): Simon Thomas

PI Email Address: cherlin@math.rutgers.edu

Abstract: Cherlin and Thomas will pursue interactions of the techniques of logic with problems in algebra and combinatorics. Cherlin will work with groups of finite Morley rank using methods modeled heavily on finite group theory, aiming particularly at an approach to the odd characteristic case compatible with the existence of bad fields, and on problems in graph theory susceptible to model theoretic analysis (universal graphs and problems of wqo). Thomas will work on Borel equivalence relations, particularly with those associated with natural classification problems in algebra, which may well provide the examples needed to settle some problems presently open in full generality, as well as providing information on the relative difficulty (according to a very robust system of measurement) of the algebraic problems, some very classical and open, for what can now seen to be essential reasons. Thomas will also pursue his work on the automorphism tower problem, using set theoretic techniques.

Mathematical logic provides tools of great generality which can be applied to various areas of mathematics. In combinatorial contexts the model theoretic point of view provides methods that can be used to handle specific problems very uniformly, rather than on the case by case basis sometimes encountered in the literature. Descriptive set theory provides methods for analyzing the relative difficulty of both solved and unsolved problems in algebra, and in particular provides concrete information as to how detailed an answer one may usefully seek in a classification problem, making it possible to distinguish dead ends from fruitful lines of inquiry on an a priori basis.