Topology, Geometry and Physics

摘要奖：DMS-0405143主要研究者：Clifford H. Taubes对C. H. Taubes将主要集中在微分拓扑学，但也有三个数学物理项目的兴趣。 主要项目包括使用辛几何工具来研究四维空间的微分拓扑。 长期的目标是阐明光滑四维空间的分类，这是低维拓扑学中的一个突出的开放问题。 由于有相当多的间接证据证明辛几何在故事中的作用，目前的计划是探索辛几何与这一主题的联系。 第一个数学物理项目的目的是构造一个等变Fredholm设置的Dirac算子的循环空间的流形。 这样的设置将为弦理论物理学的各个方面提供一个严格的框架。 第二个数学物理项目将研究量子色动力学中强耦合哈密顿量的Hartree-Fock近似的最小能态。 第三个数学物理项目是与一位化学家联合进行的;它研究了盐溶液中DNA分子的自由能。 这个项目的目的是预测这些分子在不同生物学背景下的行为，主要项目是探索四维宇宙大尺度结构的可能性。 例如，我们的宇宙有四维，这些通常的三维空间加上时间作为第四维;天文学家已经证实了爱因斯坦的预言，我们的宇宙在大尺度上是弯曲的。 这项研究试图为四维宇宙提供一个可能的大尺度结构的完整列表。 到目前为止，还没有可信的可能性列表，但有其格式的线索。 还有三个其他的研究课题是由数学物理中的问题激发的。 其中第一个试图证实物理学中统一场论中引力和亚原子力的各种假设。 第二个主题探讨了预测将亚核夸克束缚为介子、中子和质子的强力性质的技术。 第三个主题研究物理化学方程，预测DNA分子在各种生物学背景下的行为。 我们的目标是开发工具来简化计算。

AbstractAward: DMS-0405143Principal Investigator: Clifford H. TaubesThe research of C. H. Taubes will mostly concentrate in differentialtopology, but there are three mathematical physics projects ofinterest as well. The primary project involves the use of symplecticgeometry tools to investigate the differential topology of fourdimensional spaces. The long range goal is to shed light on theclassification for smooth, four dimensional spaces, this one of theoutstanding open problems in low dimensional topology. As there isconsiderable circumstantial evidence for a symplectic geometry role inthe story, the immediate plan is to probe the symplectic geometryconnections to the subject. The first of the mathematical physicsprojects aims to construct an equivariant Fredholm setting for a Diracoperator on the loop space of a manifold. Such a setting wouldprovide a rigorous framework for various aspects of string-theoreticphysics. The second mathematical physics project will study theminimal energy states for a Hartree-Fock approximation to the strongcoupling Hamiltonian for quantum chromodynamics. The thirdmathematical physics project is undertaken jointly with a chemist; itstudies a version of free energy for a DNA molecule in a saltsolution. The aim here is to predict the behavior of such moleculesin various biological contexts.The major project probes the possibilities for the large scalestructure of four dimensional universes. For example, our universehas four dimensions, these the usual three dimensions of space plustime as the fourth; and astronomers have confirmed Einsteinsprediction that our universe is curved at large scales. The researchseeks to provide a complete list of the possible large scalestructures for a four dimensional universe. As of now, there is nocredible conjectured list for the possibilities, but there are cluesto its format. There are three other research topics that aremotivated by questions in mathematical physics. The first of theseseeks to confirm various conjectures from unified field theories inphysics for gravity and the subatomic forces. The second topicexplores techniques for predicting properties of the strong force thatbinds subnuclear quarks as mesons, neutrons and protons. The thirdtopic studies equations from physical chemistry that predict thebehavior of DNA molecules in various biological contexts. The goal isto develop tools to simplify the calculations.