Topics in Formal Geometry

摘要提案：DMS-9804833主要研究者：Ian Anderson和Mark Fels形式微分几何是研究在给定微分关系的解空间上定义的局部形变不变量。这些变形不变量以上同调类的形式出现在定义在给定微分关系的射流空间上的各种不变变分双复上。本研究计划的目标是发展计算这些局部变形不变量的新方法，利用射流束技术研究这些不变量的性质，并开发形式几何在全局分析、可积系统和数学物理中的新应用。提出了三种研究途径。第一个项目将探索两个微分方程的变分复合体之间的关系，这两个方程是由各种积分或约简方法中的一种联系起来的，其中一些可以追溯到Lie和Vessiot。第二个项目是基于变分反问题的两个新的扩展。一个扩展是针对哈密顿结构的存在和分类；另一种扩展试图描述全局分析中出现的积分不变量，如阻碍规定曲率问题解决的Kazdan-Warner积分。第三个项目的动机是变分复合体在特征类的一般理论中所起的作用，并试图利用移动框架和微分不变理论的方法将这一作用推广到纤维束上的一般伪群作用。这幅作品的主题是对称。一个物体的对称性是指在空间中使该物体保持不变的一组运动。立方体的对称性包括围绕其3个轴的每一个轴进行90度的离散旋转。球体的对称群包括沿球体中心绕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解析和定性特征都是极其重要的。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。然而，还有许多理论和实践问题有待研究。特别是，以前无法通过计算实现的理论技术，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.