Cycles, Nonlinear Differential Equations, and Geometric Pluripotential Theory

摘要奖：DMS 1301804，主要研究者：H.布莱恩·劳森（Blaine Lawson）这些项目涉及周期、非线性偏微分方程和多能理论的几何推广的研究。 第一个项目涉及黎曼几何中的完全非线性微分方程。主要研究者和R.哈维的狄利克雷问题将继续下去，和问题有关的奇异性和切线的解决方案将进行调查。 这项研究的动机来自于研究人员在校准几何和其他几何中对多能理论的发展，其中引入了pluisubharmonic函数，伪凸域，容量等概念，并建立了许多基本性质。 这应该对校准几何产生重要影响，而校准几何又在现代物理学的M理论中发挥重要作用。 也应该有应用辛几何和p-凸黎曼几何。 第二部分的建议涉及的团体的代数圈和cocycle上的一个投影代数品种。 这里的目的是了解这些群体，并将它们与品种本身的全球结构联系起来。 研究者与他人建立了基于圈空间同伦群的代数簇的同调型理论。 这一理论将适用于具体问题的代数空间，并为真实的代数几何的影响将进行探讨。 也将探讨在先前的研究中出现的拓扑学中的普遍结构的惊人连接。 一个强烈相关的一部分，建议关注的周期约束全纯链在投影流形。我们将研究投射联结数和拟多重次调和函数的刻画。 这将涉及分析结构的投影外壳，一个类似于多项式外壳的概念，这已经介绍了调查员，是独立的利益。 射影壳与逼近理论、多能理论和Banach分次代数的谱有关。 最后一个领域涉及分析方法的差异字符，他们的推广，开发的PI。 中心对象在循环和平滑数据之间进行调解。在复范畴中，这涉及到德利涅上同调的分析研究。 它产生不变量的丛和叶状，并检索经典的Abel-Jacobi映射。 变分方法将承担的研究，并将其连接到1-拉普拉斯算子和极小超曲面的理论。 几何学中一个重要的概念是“循环”。在代数几何中，一个循环对应于一个多项式方程组的联立解。在微分几何中，循环以多种方式出现：作为某些微分方程的大规模解，以及作为可微映射的水平集和奇点集。空间中的曲线和曲面是简单的例子。 具有特定几何形状的周期（由研究人员及其合作者发现）在现代物理理论中发挥着重要作用。 学生将成为研究团队的一部分。也将有一个本科教育的努力，旨在培养数学的独立性。

AbstractAward: DMS 1301804, Principal Investigator: H. Blaine LawsonThese projects are concerned with the study of cycles, nonlinear partial differential equations, and geometric generalizations of pluripotential theory. The first project concerns fully nonlinear differential equations in riemannian geometry. Recent work of the principal investigator and R. Harvey on the Dirichlet problem will be continued, and questions concerning singularities and tangents to solutions will be investigated. Motivation for this study came from the investigators' development of pluripotential theory in calibrated and other geometries, where notions of plurisubharmonic functions, pseudo-convex domains, capacity, etc. were introduced and many basic properties established. This should have an important impact in calibrated geometry, which in turn plays an important role in M-theory in modern physics. There should also be applications to symplectic geometry and to p-convexity in Riemannian geometry. The second part of the proposal concerns the groups of algebraic cycles and cocycles on a projective algebraic variety. Here the aim is to understand these groups and relate them to the global structure of the variety itself. The investigator has, with others, established a theory of homology type for algebraic varieties based on the homotopy groups of cycles spaces. This theory will be applied to concrete questions about algebraic spaces, and implications for real algebraic geometry will be explored. Striking connections to universal constructions in topology which emerged in prior research will also be probed. A strongly related part of the proposal concerns cycles which bound holomorphic chains in projective manifolds. Characterizations in terms of projective linking numbers and quasi-plurisubharmonic functions will be studied. This will involve analyzing the structure of projective hulls, a concept analogous to polynomial hulls, which has been introduced by the investigator and is of independent interest. Projective hulls are related to approximation theory, pluripotential theory, and the spectrum of Banach graded algebras. The final area concerns analytic approaches to differential characters, and their generalizations, developed by the PI. The central objects mediate between cycles and smooth data. In the complex category this involves an analytic study of Deligne cohomology. It yields invariants for bundles and foliations, and retrieves the classical Abel-Jacobi mappings. Variational methods will be brought to bear on the study and connect it to the 1-Laplacian and the theory of minimal hypersurfaces. A concept of central importance in geometry is that of a "cycle." In algebraic geometry a cycle corresponds to the simultaneous solution of a system of polynomial equations. In differential geometry cycles arise in many ways: as the large scale solutions of certain differential equations, and as the level sets and singularity sets of differentiable mappings. Curves and surfaces in space are simple examples. Cycles with a particular geometry (discovered by the investigator and his collaborators) play a fundamental role in modern physical theories This project will also be concerned with graduate student development. Students will be part of the research team. There will also be an undergraduate educational effort aimed at fostering mathematical independence.