The investigator addresses problems in liquid-crystalline and bacterial systems related to formation, evolution, and interaction of singularities and defects. Such singularities manifest themselves in mathematical equations when the underlying physical models lose validity on a given scale. To study such systems, one must connect often quite disparate physical models arising at different length and time scales. The investigator employs a combination of modern analytical and numerical methods introduced by him for the Onsager model of nematic liquid crystals and Keller-Segel model of bacterial chemotaxis. In particular, he investigates liquid-crystalline systems such as polydisperse and biaxial nematics, smectics, and elastomers; phenomenon of formation and interaction of particle aggregates in bacterial chemotaxis. He creates the Multiphysics Modeling Lab: a center for research and education in methods of mathematical modeling and computer simulations. This establishes a collaborative environment involving students at the University of Arizona as well as (utilizing the close connection of the investigator with the Tucson Math Circle) the local K-12 students in projects on modeling and simulation of complex systems.
The fundamental significance of the areas of this project has been widely recognized, e.g., by a Nobel Prize awarded in 1991 to P.-G. de Gennes for fundamental contributions to our understanding of order phenomena in complex systems, and by a Microsoft Award in 2006 to D. Bray for his work on chemotaxis on E. coli. The practical applications of the project range from liquid-crystalline display (LCD) technology to understanding the functioning of our immune system. Recently, the analytical and numerical methods employed in the investigations of such systems have matured enough to allow for new significant advances in these fields. Achieving this is the primary scientific goal of this project. Beyond that, the interdisciplinary nature of this program is perfect for illustrating the power of mathematics in describing diverse natural phenomena and is suitable for involving students of all levels. The Multiphysics Modeling Lab involves graduate and undergraduate students in active applied mathematics research and promotes mathematics and science among the K-12 students and educators via close integration with the system of math circles. Such activities are essential for improving the general level of scientific education in our society.
