Spectra, gaps, degenerations and cycles

PI将继续他们在同调镜像对称（HMS）方面的基础工作，发展HMS中涉及的结构和理论，并建立在这些理论的应用基础上。一个值得注意的方向是理论的高级辛结构，它带来了“堆叠”的方向和“导出”的方向之间的对偶性的大多数模问题在代数几何。要充分利用这些结构的深度，就需要仔细研究各种范畴实体和更高范畴实体的模。这是所有首席执行官的主要专业领域之一。Kontsevich介绍了一个主要的工具，派生计划。 Katzarkov提出了LG模型的模及其单值性可以被解释为稳定性条件和谱的想法。这些活动适合于一个更普遍和全球性的哲学，旨在伴随几何学在21世纪。世纪的几何学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，这些对象，各种各样的，被一致地视为某种“点集”。沿着这条路，与范畴结构的关系稳步增长，导致我们的程序中有许多输入，如上所述。在世纪之交取得的许多进步中，PI本身发挥了关键作用。随着PI Kontsevich对HMS的介绍，一个微妙的变化被引入，在“几何”中开始被视为一个范畴结构。与此同时，更高层次理论的发展意味着几何结构不再被视为“点的集合”，而是被视为包围更高层次结构的对象。该项目与理论物理学密切相关。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，这场革命将由欧洲核子研究中心（CERN）大型强子对撞机（LHC）的新实验结果带来。这些将有助于确定目前开放的众多理论可能性中的哪一个，最好地解决高能尺度下的量子场论。对于这些理论，要知道哪些是正确的参数。因此，在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。 随着HMS和超对称理论之间的关系，随着更高范畴和TQFT之间的关系，随着配分函数和非交换上同调之间的关系，我们将在这个项目中研究的几何对象的种类对于理解理论物理中的这些新的几何对象变得至关重要。该项目有一个教育组成部分-会议和教育博士后，这个组成部分在过去取得了巨大的成功，随着更多的资金，我们计划把它带到一个新的水平。

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.