Topology, Geometry and Physics

主要研究者：Clifford H. taubesc . H. taubesc . H. taubesc . H. taubesc . H. taubesc . H. Taubes的研究主要集中在微分拓扑上，但也有三个感兴趣的数学物理项目。主要项目涉及使用辛几何工具来研究四维空间的微分拓扑。长期目标是阐明光滑四维空间的分类，这是低维拓扑中突出的开放问题之一。由于有相当多的间接证据表明辛几何在故事中扮演着重要的角色，因此我们的直接计划是探索辛几何与主题的联系。第一个数学物理项目旨在为流形的循环空间上的Diracoperator构造一个等变Fredholm集。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与一名化学家共同完成；它研究了盐溶液中DNA分子的自由能。这里的目的是预测这些分子在各种生物环境中的行为。主要项目是探索四维宇宙的大规模结构的可能性。例如，我们的宇宙有四个维度，这些通常是空间的三个维度加上时间作为第四个维度；天文学家证实了爱因斯坦的预言，即我们的宇宙在大尺度上是弯曲的。这项研究旨在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，有一个令人难以置信的猜测列表，但它的格式有一些线索。还有三个其他的研究课题是由数学物理问题激发的。第一个是试图证实物理学中关于引力和亚原子力的统一场论的各种猜想。第二个主题探讨了预测将亚核夸克作为介子、中子和质子结合在一起的强作用力性质的技术。第三个主题研究物理化学方程，预测DNA分子在各种生物环境中的行为。目标是开发简化计算的工具。

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.