Spectra, gaps, degenerations and cycles

光谱、间隙、简并和循环

• 批准号: 1201475
• 负责人: Ludmil Katzarkov
• 金额: $ 24.3万
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201475&HistoricalAwards=false
• 关键词: Spectra; gaps; degenerations; cycles

The PIs will continue their foundational work on Homological Mirror Symmetry (HMS), to develop the structures and theories involved in HMS, and to build on applications of these theories. One notable direction is the theory of higher symplectic structures, which brings out the duality between the ``stacky'' directions and the ``derived'' directions of most moduli problems in algebraic geometry. Exploiting the full depth of these structures will require a careful study of the moduli of various kinds of categorical and higher categorical entities. This is one of the main areas of expertise of all the PIs. Kontsevich introduced one of the main tools, derived schemes. Katzarkov came up with the idea that moduli of LG models and its monodromy can be interpreted as stability conditions and spectra.These activities fit into a more general and global philosophy designed to accompany Geometry in the 21st Century. The study of Geometry in the 20th Century was devoted, in large part and with astounding success, to the classification and parametrization of geometrical objects. However,these objects, of various kinds, were uniformly viewed somehow as ``sets of points''. Along the way, the relationship with categorical structures grew steadily, leading to the many inputs into our program as discussed above. The PIs themselves played a pivotal role in much of the progress that was made at the turn of the century. With PI Kontsevich's introduction of HMS, a subtle change was introduced, in that ``Geometry'' began to be seen within a categorical structure. And the concurrent development of the theory of higher stacks meant that geometric structures were no longer viewed just as ``sets of points'' but rather as objects enclosing a higher structure. This project is highly connected with theoretical physics. As we head into the second decade of the 21st Century, elementary particle physics is on the crux of a profound revolution to be brought about by the new experimental results coming out of the LHC at CERN. These will serve to identify which of the multitude of theoretical possibilities which are currently open, best address quantum field theory at the high energy scale. And for those theories, to tell which are the right parameters. So there will soon be a lot of work to do on the theoretical side, and this will surely require new tools and a new approach. With the relationship between HMS and supersymmetric theories, with the relationship between higher categories and TQFT, with the relationship between partition functions and nonabelian cohomology, the kinds of geometrical objects which we are going to investigate in this project are becoming crucial for understanding these new panoramas in theoretical physics. The project has an educational component - conferences and educating postdocs, This component has been hugely successful in the past and with more funding we plan to bring it to the next level.

PI将继续其在同源镜对称性（HMS）上的基础工作，以发展涉及HMS的结构和理论，并基于这些理论的应用。一个值得注意的方向是较高的符号结构的理论，它提出了``stacky''方向与代数几何形状中大多数模量问题的``stacky''方向之间的双重性。利用这些结构的全部深度将需要仔细研究各种类型和更高分类实体的模量。这是所有PI的专业知识的主要领域之一。 Kontsevich介绍了主要工具之一，即派生的方案。 Katzarkov提出了这样的想法，即LG模型的模量及其单片可以解释为稳定条件和频谱。这些活动适合于21世纪旨在伴随几何形状的更一般和全球哲学。在20世纪的几何学研究很大程度上是对几何物体的分类和参数化的很大程度上的。但是，这些对象各种物体被以某种方式统一视为``点''。在此过程中，与分类结构的关系稳步增长，导致了上面讨论的许多投入。 PI本身在世纪初取得的许多进步中发挥了关键作用。随着Pi Kontsevich对HMS的介绍，引入了微妙的变化，因为“几何形状”开始在类别结构中看到。高等堆栈理论的并发发展意味着几何结构不再被视为``点'''，而是作为包含更高结构的对象。该项目与理论物理学高度联系。当我们进入21世纪第二个十年时，基本粒子物理学正是由CERN的LHC出现的新实验结果带来的深刻革命的症结所在。这些将有助于确定当前开放的多种理论可能性，最佳的量子场理论在高能量尺度上。对于这些理论，要确定哪个是正确的参数。因此，很快就会在理论方面做很多工作要做，这肯定需要新的工具和新的方法。 与HMS与超对称理论之间的关系以及更高类别与TQFT之间的关系以及分区函数与非亚伯群岛的关系之间的关系，我们将在该项目中要研究的几何对象类型对于理解理论物理学中的这些新的全景。该项目有一个教育组成部分 - 会议和教育博士后，该组成部分在过去取得了巨大成功，并有更多的资金将其提升到一个新的水平。