Topics in Formal Geometry

建议：DMS-9804833主要研究人员：Ian Anderson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域形变不变量。这些形变不变量是定义在给定微分关系的喷射空间上的各种不变变分双复形中的上同调类。这项研究的目的是开发计算这些局域形变不变量的新方法，通过使用喷束技术研究这些不变量的性质，并开发形式几何在整体分析、可积系统和数学物理中的新应用。提出了研究的三条途径。第一个项目将探索两个微分方程的变分双复形之间的关系，这两个方程通过不同的积分或归化方法之一联系在一起，其中一些方法可以追溯到Lie和Vessiot。第二个项目是基于变分反问题的两个新的扩展。一种推广针对哈密顿结构的存在和分类；另一种推广试图刻画整体分析中出现的积分不变量，如Kazdan-Warner积分，它阻碍了指定曲率问题的解决。第三个项目的动机是变分双复数在一般特征类理论中的作用，并试图利用移动标架方法和微分不变量理论将这种作用推广到纤维丛上的一般伪群作用。提议的作品的流行主题是对称。所谓物体的对称性，是指在空间中保持物体不变的一组运动。立方体的对称性包括围绕它的三个轴中的每一个旋转90度。球体的对称群包括围绕通过球心的任意轴的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，人们对保持控制(微分)方程不变的对称性感兴趣。这种对称性在研究控制方程的分析和定性特征时是极其重要的。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，并且可以通过对称方法获得。尽管如此，仍有许多理论和实践问题有待研究。特别是，以前在计算上无法获得的理论技术现在可以使用强大的符号数学计算机程序来研究。

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.