This program concerns the behavior of systems with many degrees of freedom - either infinite or asymptotically large. Under these circumstances, the detailed behavior of all the constituents is essentially a meaningless concept and one looks for (a) collective behavior of the system as a whole or (b) a statistical description of the behavior of an individual component. The analyses of various systems pursued by the PI will, whenever feasible, embody both descriptions. The studies will use four interrelated methods from depictions of condensed matter problems. These are (i) Classical Spin & Particle Systems, (ii) Quantum Spin Systems, (iii) Interacting Particle Models and (iv) Cluster Models/Graphical Representations. Problems from all four categories will be examined. Specific instances include discontinuous phase transitions in quantum, classical and continuum fluid systems, the disruptions and enhancements of transport due to inhomogeneities in disordered media and the approach of critical 2D systems to their continuum conformal invariant limits. The PI maintains extensive contact with physicists in the areas of field theory and condensed matter physics as well as mathematicians from various subdisciplines. Most often the PI brings to bear the necessary mathematics in order to elicit enhanced understanding of a physical system. On occasion, physical insight for a particular system can influence the associated areas of mathematics.
Broader Impact. A philosophy which became prevalent beginning in the late 1960s is that of universality - largescale and/or longtime collective behaviors do not depend on the details of the model. Hence, while the primary scientific goals of this proposal lean towards the advancement of the physical sciences, these investigations lead directly to considerations of models with supplemental applications. In particular: (1) Studies of collective swarming behavior in biological systems and in crowded human aggregations with further applications to surveillance and gathering of intelligence. (2) Studies of criminal behavior with applications to hotspot formation in the statistical occurrences of residential burglaries and to hostility reciprocations of rival street gangs. (3) Studies of the dissemination and flow of information in large scale networks. This research encompasses an extensive array of applications research opportunities at every level. The program will require the assistance of many collaborators and extensive use of the facilities at UCLA. The PI has collaborated with undergraduate, graduate, postdoctoral, and senior level coresearchers from a variety of cultural backgrounds with both genders adequately represented.
