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 将继续他们在同源镜像对称 (HMS) 方面的基础工作，开发 HMS 涉及的结构和理论，并以这些理论的应用为基础。一个值得注意的方向是高辛结构理论，它提出了代数几何中大多数模问题的“堆叠”方向和“导出”方向之间的对偶性。充分利用这些结构的深度将需要仔细研究各种分类和更高分类实体的模。这是所有 PI 的主要专业领域之一。 Kontsevich 介绍了主要工具之一：派生方案。 Katzarkov 提出了 LG 模型的模及其一律可以解释为稳定性条件和谱的想法。这些活动符合旨在伴随 21 世纪几何的更普遍和全球性的哲学。 20 世纪的几何研究在很大程度上致力于几何对象的分类和参数化，并取得了惊人的成功。然而，这些不同种类的物体在某种程度上都被统一视为“点集”。一路走来，与分类结构的关系稳步增长，导致我们的程序得到了如上所述的许多输入。 PI 本身在世纪之交取得的许多进展中发挥了关键作用。随着 PI Kontsevich 引入 HMS，引入了微妙的变化，“几何”开始在分类结构中被看到。更高层次理论的同时发展意味着几何结构不再仅仅被视为“点集”，而是被视为包围更高结构的对象。该项目与理论物理密切相关。当我们进入 21 世纪的第二个十年时，基本粒子物理学正处于一场深刻革命的关键，欧洲核子研究中心大型强子对撞机的新实验结果将带来这场革命。这些将有助于确定目前开放的众多理论可能性中哪些最能解决高能尺度的量子场论。对于这些理论，要判断哪些是正确的参数。因此，理论方面很快就会有很多工作要做，这肯定需要新的工具和新的方法。 通过HMS和超对称理论之间的关系，通过高范畴和TQFT之间的关系，通过配分函数和非阿贝尔上同调之间的关系，我们在这个项目中要研究的几何对象的种类对于理解这些新的全景图变得至关重要在理论物理学中。该项目有一个教育部分——会议和教育博士后，这个部分在过去取得了巨大的成功，我们计划通过更多的资金将其提升到一个新的水平。