Geometric PDE in Confromal Geometry and Relativity

项目负责人：Jie Qing （University of California, Santa Cruz）题目：保形几何和相对论中的几何PDE杰青提出研究共形紧致爱因斯坦流形。共形紧致爱因斯坦流形在所谓的AdS/CFT对应中起着至关重要的作用，这是一个非常有前途的理论，它预测了量子引力理论和某些共形场论之间的某些联系。在数学中，人们已经知道，共形紧致爱因斯坦流形为研究共形几何提供了一个强有力的途径。散射矩阵研究的最新进展为理解物理学中的全息原理提供了一种较为全面、自然的方法，这是非常令人着迷和有希望的。Jie Qing博士对渐近ad /dS时空理论的发展和研究也很感兴趣。首先，从统计学角度考虑AdS/dS空间时间。这个理论不像渐近平坦时空的研究那样有名。但它在经典相对论和量子引力理论中同样重要，如果不是更重要的话。提出的研究是研究全息原理理论中的主要对象，涉及量子引力理论和一些共形场理论。它已经成为数学家和物理学家可以互动的领域的一部分。它是为所有基本相互作用（包括强弱核相互作用、电磁和引力）寻找统一理论的努力的一部分。这一领域的进步将大大提高我们对自然界的认识。本研究是加州大学圣克鲁斯分校数学系本科和研究生教育项目的重要组成部分。

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.