Topics in Formal Geometry

摘要 提案：DMS-9804833主要研究者：Ian安德森和Mark Fels 形式微分几何是研究定义在给定微分关系的解空间上的局部变形不变量。 这些变形不变量出现的上同调类在各种不变的变分双复定义的射流空间的给定的微分关系。 这项研究的目的是开发新的方法来计算这些局部变形不变量，研究这些不变量的性质，通过使用射流束技术，并开发新的应用程序的形式几何在全球分析，可积系统和数学物理。 提出了三种研究途径。 第一个项目将探讨两个微分方程的变分双复形之间的关系，这两个微分方程由各种积分或简化方法之一相关，其中一些方法可以追溯到Lie和Vessiot。 第二个项目是基于两个新的扩展的反问题的变分法。 一个扩展是针对汉密尔顿结构的存在和分类;另一个扩展旨在描述在全球分析中产生的积分不变量，如卡兹丹-华纳积分，这阻碍了解决规定的曲率问题。 第三个项目的动机的作用，变分bicomplex发挥的一般理论的特征类，并试图推广这一作用，一般伪群作用的纤维丛使用的方法移动框架和微分不变理论。 拟议中的工作的普遍主题是对称性。 所谓物体的对称性，是指在空间中使物体保持不变的一组运动。 立方体的对称性包括围绕其3个轴中的每一个轴90度的离散旋转。 球面的对称群包括绕通过球心的任意轴作任意角度的连续旋转。 在应用数学、微分几何和数学物理的问题中，人们感兴趣的是保持控制（微分）方程不变的对称性。 这种对称性对于研究控制方程的分析和定性特征都极其重要。 事实上，几乎所有已知的重要非线性方程（如爱因斯坦方程、杨-米尔斯方程或调和映射方程）的精确解都具有高度的对称性，并且可以通过对称性方法获得。 然而，还有许多理论和实践问题有待研究。 特别是，以前无法计算的理论技术现在可以使用强大的符号数学计算机程序进行研究。

Abstract Proposal: DMS-9804833 Principal Investigators: Ian Anderson and Mark Fels Formal differential geometry is the study of localized deformation invariants defined on the solution space of a given differential relation. These deformation invariants arise as cohomology classes in various invariant variational bicomplexes defined on the jet spaces of the given differential relation. The objectives of this research proposal are to develop new methods for the computation of these localized deformation invariants, to study the properties of these invariants through the use of jet bundle techniques, and to develop new applications of formal geometry in global analysis, integrable systems and mathematical physics. Three avenues of research are proposed. The first project will explore the relationships between the variational bicomplex for two differential equations which are related by one of the various integration or reduction methods, some of which date back to Lie and Vessiot. The second project is based upon two new extensions of the inverse problem of the calculus of variations. One extension is directed towards the existence and classification of Hamiltonian structures; the other extension seeks to characterize integral invariants arising in global analysis such as the Kazdan-Warner integrals which obstruct the solution to the prescribed curvature problem. The third project is motivated by the role that the variational bicomplex plays in the general theory of characteristic classes and seeks to generalize this role to general pseudo-group actions on fiber bundles using the method of moving frames and differential invariant theory. The prevalent theme of the proposed work is symmetry. By the symmetry of an object one means the group of motions in space which leave the object unchanged. The symmetry of a cube includes discrete rotations by 90 degrees about each one of its 3 axis. The symmetry group of a sphere includes the continuous rotations t hrough any angle about any axis through the center of the sphere. In problems in applied mathematics, differential geometry and mathematical physics, one is interested in the symmetries which leave the governing (differential) equations unchanged. Such symmetries are extremely important in studying both the analytical and qualitative features of the governing equations. Indeed, almost all known exact solutions of important non-linear equations (such as the Einstein, Yang-Mills, or harmonic map equations) have a high degree of symmetry and can be obtained by symmetry methods. Nevertheless, there are many theoretical and practical issues yet to be studied. In particular, theoretical techniques which have previously been computationally inaccessible can now be investigated using powerful symbolic mathematics computer programs.