Geometric PDE in Confromal Geometry and Relativity

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Proposal DMS-0402294Principal Investigator: Jie Qing (University of California, Santa Cruz)Title: Geometric PDE in conformal geometry and relativityABSTRACTDr. Jie Qing proposes to study conformally compact Einstein manifolds. Conformally compact Einstein manifolds play crucial roles in theso-called AdS/CFT correspondence, which is a very promising theory that predicts certain connections between quantum gravitational theory andcertain conformal field theory. In mathematics it has been knownthat conformally compact Einstein manifolds provide a powerful approach tothe study of conformal geometry. Recent development in the study of thescattering matrix in providing rather global, natural way to understandthe holography principle in physics seems very fascinating and promising. Dr. Jie Qing is also interested in developing and study theory aboutasymptotically AdS/dS space-times. For a start, one considers staticasymptotically AdS/dS space-times. The theory is not as well-known as thestudy of asymptotically flat space-times. But it is as significant, if notmore, in the classic relativity and quantum gravitation theory.The proposed research is to study the primary object in thetheory of holography principles that relate quantum gravitation theory and some conformal field theory. It has become a part the field where mathematicians and physicists can interact. It is a part ofthe efforts to find a unified theory for all fundamental interactionsincluding the strong and weak nuclear interactions, electromagnetism, andgravity. Advancements in this field will greatly improve our understandingof the nature in theory. The proposed research is a very important part ofthe undergraduate and graduate education programs in the department ofmathematics at University of California, Santa Cruz.