NSF Award
0854779
The most challenging problem in theoretical physics is the unification of the laws of quantum physics with Einstein's general theory of relativity. The ideas in this direction that have been pursued most forcefully are superstring theory and noncommutative geometry. These two approaches are intimately related. Noncommutative geometry has provided a new framework to study spaces which could not be handled by the standard methods of Riemannian geometry. It provides a new perspective in the study of gauge theories and is a promising candidate to describe geometries at Planckian energies. The main focus of this proposal is to study the nature of space-time at subplanckian scales, determining the dynamics and finding connections between superstring theory and noncommutative geometry. It is essential to find the noncommutative space-time whose limit is the product of a continuous four-dimensional manifold times a finite space that reproduces the standard model. Also to be invesigated will be the functional integral for chiral Dirac operators in the random matrix approximation, deforming space-time with continuous parameters, heat kernel methods to find boundary terms for the spectral action, adding an inflaton, dilaton and axion, the behavior of spectral actions in presence of singularities and nonperturbative evolution of the dynamical fields at scales of the order of the planck length. Topological five-dimensional gravity, regular BPS solutions with cylindrical symmetry, and exact solutions for the bigravity system will be studied. <br/>Broader impact : Due to its interdisciplinary nature, this work, on the frontiers of noncommutative geometry and string theory, will be of benefit to both physicists and mathematicians. The American University of Beirut (AUB) plays a very important role in promoting the American educational system in the Middle East. Historically, the emphasis was on liberal education. The new aim is to provide in addition a vibrant environment for the promotion of research. The launching of the Ph.D. program in theoretical physics in October 2008 is a step in this direction and will hopefully attract students of high caliber from other universities in the Middle East.
