Space-Time from the Spectral Point of View: Towards Quantum Gravity

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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 offered a new perspective in the study of gauge theories and the role of the Higgs field and is a promising candidate to describe geometries at Planckian energies. In the past years it has been established that space-time is approximated by a well defined product of a continuous four-dimensional manifold by a finite dimensional space. The main focus of this program is to construct a genuinely noncommutative finite dimensional space valid at sub-planckian scales and whose limit at lower energies is the noncommutative space of the standard model. The geometry will be reduced to those of almost commuting matrices, and it will be possible to get a handle on quantizing the geometry. It will also study new asymptotic limits of the spectral action as well as cosmological implications. Other parts of this project will study various aspects of gravity such as stability of BPS monopoles, and formulation of consistent massive theories of gravity, implications of the presence of right-handed neutrinos on the geometry of spacetime, the behavior of spectral actions in the presence of singularities and non-perturbative evolution of the dynamical fields at scales of the order of the Planck length. Broader impact of Project: Due to its interdisciplinary nature, this work which involves research on the frontiers of noncommutative geometry, will be of benefit to both physicists and mathematicians. The new ideas and methods in this proposal will contribute to the knowledge of the high energy theory and phenomenology, quantum gravity and mathematics communities, and to a lesser extent to the astro-physics community. 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 2009 is a step in that direction and is attracting students of high caliber.