Cycles, Nonlinear Differential Equations, and Geometric Pluripotential Theory

摘要奖：DMS 1301804，首席研究员：H.布莱恩·劳森这些项目涉及循环、非线性偏微分方程和多势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。首席研究员和R.Harvey最近关于狄利克莱特问题的工作将继续进行，并将研究有关奇点和解的切线的问题。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发展的多势理论，其中引入了多重亚调和函数、伪凸域、容量等概念，并建立了许多基本性质。这应该对校准几何学产生重要影响，而校准几何学反过来又在现代物理学中的M理论中扮演着重要角色。也应该应用于辛几何和黎曼几何中的p-凸性。该建议的第二部分涉及射影代数簇上的代数圈群和余圈群。这里的目的是了解这些群体，并将它们与多样性本身的全球结构联系起来。在圈空间同伦群的基础上，与其他人一起建立了代数簇的同调型理论。这一理论将被应用于关于代数空间的具体问题，并将探索对真实代数几何的影响。在拓扑学中出现的与普遍结构的惊人联系也将被探索。该建议的一个强相关部分涉及到射影流形中约束全纯链的圈。我们将研究用射影连接数和拟多重亚调和函数的刻画。这将涉及到分析射影壳的结构，这是一个类似于多项式壳的概念，已由研究人员引入，并具有独立的兴趣。射影壳与逼近理论、多位势理论以及Banach分次代数的谱有关。最后一个领域涉及到由PI开发的对差异特征及其泛化的分析方法。中心对象在循环和平滑数据之间进行调节。在复数范畴中，这涉及到Deligne上同调的分析研究。它产生丛和叶层的不变量，并检索经典的Abel-Jacobi映射。变分方法将应用于研究，并将其与1-拉普拉斯理论和极小超曲面理论联系起来。几何学中最重要的一个概念是“循环”。在代数几何中，循环对应于多项式方程组的同时解。在微分几何中，循环以多种方式出现：作为某些微分方程解的大规模解，以及作为可微映射的水平集和奇异集。空间中的曲线和曲面就是简单的例子。具有特定几何形状的循环(由研究人员和他的合作者发现)在现代物理理论中扮演着基本的角色，这个项目也将关注研究生的发展。学生将成为研究团队的一部分。还将有一项旨在培养数学独立性的本科教育努力。

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.