Moving Frames on Lattices and Applications

格子上的移动框架及其应用

• 批准号: 1405722
• 负责人: Gloria Mari-Beffa
• 金额: $ 17.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405722&HistoricalAwards=false
• 关键词: Moving; Frames; Lattices; Applications

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.

摘要奖：DMS 1405722，首席研究员：Gloria Mari-Beffa完全可积方程的解具有真正显著的性质。一个常见的例子是浅水中传播的孤立波的运动：尽管波往往是以家庭形式传播的，但在浅水中，人们可以观察到孤立波不变地传播，似乎永远不变，而且非常稳定，例如，它们不会受到与另一种此类波的正面碰撞的干扰。这些被称为“孤子”，在它们的方程中，我们有那些控制飞机尖端后面的尾涡、烟圈和气泡环等的方程。众所周知，孤子与几何有着密切的联系，当存在某种几何时，一些方程就会自然地出现。(设置适当的几何背景通常是解决问题的基本步骤，因为选择几何建立了我们希望保持不变的属性和定律；就像屏幕上的3D图像一样。)例如，当在计算机中3D图像的自然几何投影平面中工作时，曲线的某种运动将表现为行进的孤立波，而如果在通常的欧几里德平面中考虑，则不会。但图像不是连续的曲线，它们是由一组离散的像素组成的，而曲线的运动实际上是多边形的运动。现实是离散的，而不是连续的。在这个方案中，我们将研究与不同几何中的多边形的运动有关的完全可积离散系统，包括射影平面。我们将使用的工具之一是晶格上的离散运动框架，它是从晶格的顶点到顶点变化的参照系，在不变理论中有重要的作用。连续理论广为人知，但离散理论现在正在发展中。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的几何形状。在这个项目中，主要研究人员建议研究格子上离散运动框架的概念，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，以及另一方面完全可积格子系统之间的关系，包括从流的不同几何图形生成相关哈密顿结构的可能性。她还将研究离散运动标架与格点的局部差分几何的相关性，以及应用代数方法产生几何显著不变量的可能性。最后，她想要研究一个现实生活中的应用，研究分子的形状，从电子显微镜获得的图像。该提案的解决方案可能会将Cartans几何、李理论和不变量理论的部分内容带入使用几何信息潜力很大的学科。这些都是非常丰富的领域，我们正在提议开发技术，使其部分内容可以通过计算访问。完全可积偏微分方程组与曲线曲面的局部几何之间的关系已经建立，许多方面的可积性都有运动标架的演化、曲线流的摄动、Maurer-Cartan联络的拉回等几何解释。从这里直接转移到它们的离散使用差分几何不仅有趣，而且它有可能产生偏微分方程组的可积离散和格的几何相关的代数不变量等。应用问题的好处是显而易见的：如果一个问题表现出某些对称性，能够将其简化为不变量，可以降低维度，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。