Moving Frames on Lattices and Applications

摘要奖：DMS 1405722，主要研究者：格洛丽亚玛丽-贝法完全可积方程的解具有真正显着的性质。一个常见的例子是浅水沃茨中孤立波的运动：尽管波倾向于以家族的形式传播，但在浅水沃茨中，人们可以观察到孤立波，它们永远不变地传播，并且是如此稳定，以至于它们不会受到例如与另一个这样的波正面碰撞的干扰。这些被称为“孤立子”，在它们的方程中，我们有那些控制飞机尖端后面的尾涡，烟雾和气泡环，以及其他。孤立子与几何学有着密切的联系，当存在某种几何学时，一些方程会自然出现。（设置适当的几何背景通常是解决问题的基本步骤，因为几何的选择建立了我们希望保持不变的性质和定律;就像屏幕上的3D图像一样。例如，当在投影平面（计算机中3D图像的自然几何结构）中工作时，曲线的某些运动将表现为行进孤波，而如果在通常的欧几里得平面中考虑它们则不会。但是图像不是连续的曲线，它们是由一组离散的像素组成的，曲线的运动实际上是多边形的运动。现实是离散的，不是连续的。在这个建议中，我们将研究完全可积的离散系统与运动的多边形在不同的几何形状，包括投影平面。我们将使用的工具之一是格上的离散移动标架，即从格的顶点到顶点变化的参考标架，这在不变量理论中具有重要作用。连续理论已广为人知，但离散理论正在发展之中。这些框架可以应用于非常广泛的问题，包括计算和成像问题。我们的应用之一是关于使用离散框架来研究模糊图像的几何形状，例如通过Cryon电子显微镜获得的细胞图像。在这个项目中，主要研究者提出研究晶格上离散移动框架的概念，并研究其可能的应用。她想研究连续和离散版本之间的联系，以及多边形空间上的演化和不变映射之间的关系，一方面，另一方面，完全可积的晶格系统，包括从流的差分几何中可能产生相关的哈密顿结构。她还想调查的相关性离散移动框架的局部差异几何格和可能性应用代数方法产生几何显着的不变量。最后，她想从事一项真实的生活应用，从冷冻电子显微镜获得的图像中研究分子的形状。该提案的解决方案可以将Cartans几何，Lie理论和不变量理论的一部分带入使用几何信息的潜力很高的学科。这些都是非常丰富的领域，我们正在建议开发技术，使其部分计算访问。完全可积偏微分方程与曲线和曲面的局部几何之间的关系已经建立，并且可积性的许多方面都具有几何解释，如移动标架的演化，曲线流的扰动，Maurer-Cartan连接的拉回等。从这里直接移动到使用差分几何的离散化不仅有趣，但它有可能产生可积离散的偏微分方程和几何相关的代数不变量的格，等等。应用问题的优势是显而易见的：如果一个问题显示出某些对称性，能够将其简化为不变量，降低了维度，使它们更容易理解。这不仅对图像分析很重要，而且对其他数据的分析也很重要。

AbstractAward: DMS 1405722, Principal Investigator: Gloria Mari-Beffa Completely integrable equations have solutions with truly remarkable properties. A common example is the motion of traveling solitary waves in shallow waters: even though waves tend to travel in families, in shallow waters one can observe solitary waves that travel unchanged, seemly forever, and which are so stable that they are unperturbed by, for example, colliding frontally with another such wave. These are called "solitons", and among their equations we have those governing the trailing vortices behind the tips of an airplane, smoke and bubble rings, and others. Solitons are known to have close connections to geometry, with some equations appearing naturally when a certain geometry is present. (Setting an appropriate geometric background is often a fundamental step in the resolution of a problem as a choice of geometry establishes the properties and laws we wish to keep unchanged; like those of a 3D image in a screen.) For example, when working in the projective plane, the natural geometry of a 3D image in a computer, a certain motion of curves will behave as traveling solitary waves, while they would not if considered in the usual Euclidean plane. But images are not continuous curves, they are made of a discrete set of pixels, and the motion of a curve is in fact the motion of a polygon. Reality is discrete, not continuous. In this proposal we will study completely integrable discrete systems associated to motions of polygons in different geometries, including the projective plane. One of the tools we will use are discrete moving frames on lattices, frames of reference that change from vertex to vertex of the lattice, and that has an important role in invariant theory. The continuous theory is widely known, but the discrete one is now being developed. These frames can be applied to a very wide range of problems, including problems in computation and imaging. One of our applications concerns the use of discrete frames to study geometric shapes of blurry images like the ones of a cell obtained through a Cryon-electron microscope.In this project the principal investigator proposes to research the concept of a discrete moving frame on a lattices, and to investigate its possible applications. She would like to investigate the connection between continuous and discrete versions, together with the relation between evolutions and invariant maps on the space of polygons, on the one hand, and completely integrable lattice systems on the other, including the possible generation of relevant Hamiltonian structures from the difference geometry of the flow. She will also like to investigate the relevance of discrete moving frames to the local difference geometry of lattices and the possibility of applying algebraic methods to produce geometrically significant invariants. Finally, she would like to work on a real life application to study the shape of molecules from the images obtained by Cryon-electron microscopes. The resolution of the proposal could bring parts of Cartans geometry, Lie theory and invariant theory into subjects where the potential of using geometric information is high. These are very rich areas and we are proposing to develop techniques that would make parts of it computationally accessible. The relationship between completely integrable PDEs and the local geometry of curves and surfaces has already been established and many aspects of integrability have a geometric interpretation as evolutions of moving frames, perturbation of curve flows, pull back of Maurer-Cartan connections, etc. To move from here directly to their discretization using difference geometry is not only interesting, but it has the potential of producing integrable discretizations of PDEs and geometrically-relevant algebraic invariants of lattices, among others. The advantages for applied problems are clear: if a problem displays certain symmetries, being able to reduce it to its invariants lowers the dimension and makes them more accessible. This is important not only for image analysis, but also for the analysis of other data.