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，发展同调镜像对称所涉及的结构和理论，并建立这些理论的应用。一个值得注意的方向是高辛结构理论，它揭示了代数几何中大多数模问题的“堆栈”方向和“派生”方向之间的对偶性。利用这些结构的全部深度将需要仔细研究各种类别和更高类别实体的模。这是所有pi的主要专业领域之一。Kontsevich介绍了主要工具之一，衍生方案。Katzarkov提出了LG模型的模及其单一性可以解释为稳定条件和谱的想法。这些活动符合一个更普遍和全球性的哲学，旨在伴随21世纪的几何学。20世纪的几何学研究在很大程度上致力于几何物体的分类和参数化，并取得了惊人的成功。然而，这些不同种类的物体，在某种程度上被一致地视为“点的集合”。在此过程中，与范畴结构的关系稳步发展，导致了如上所述的许多输入到我们的程序中。在世纪之交取得的进步中，私人政党本身发挥了关键作用。随着PI Kontsevich引入HMS，一个微妙的变化被引入，“几何”开始在一个分类结构中被看到。与此同时，高层堆叠理论的发展意味着几何结构不再被视为“点的集合”，而是被视为包裹着高层结构的物体。这个项目与理论物理密切相关。随着我们进入21世纪的第二个十年，欧洲核子研究中心（CERN）大型强子对撞机（LHC）的新实验结果将带来一场深刻革命，基本粒子物理学正处于关键时刻。这将有助于确定目前开放的众多理论可能性中，哪一个最能解决高能量尺度的量子场论。对于这些理论，要分辨哪些是正确的参数。因此，很快就会有很多理论方面的工作要做，这肯定需要新的工具和新的方法。有了HMS和超对称理论的关系，有了高范畴和TQFT的关系，有了配分函数和非阿贝尔上同调的关系，我们在这个项目中要研究的几何对象的种类对于理解理论物理中这些新的全景是至关重要的。该项目有一个教育部分——会议和培养博士后，这部分在过去取得了巨大的成功，随着更多的资金，我们计划将其提升到一个新的水平。

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.