GRK 1821: Cohomological Methods in Geometry

这个研究项目是以几何学为基础的。我们的研究项目涵盖了从数学物理到数论的广泛学科，然而，用于研究这些不同问题的方法是密切相关的。在我们所有的项目中使用的最明显的统一技术是上同调，这是一个多功能的工具，对所有几何学科都至关重要，在未来几年中将变得更加重要。我们几乎所有的项目都使用霍奇理论，狄拉克算子，变形理论，李群或代数几何。这导致我们正在利用的协同作用。抽象代数和具体几何之间的相互作用是我们研究的主旨。参与的研究人员在几何学的不同分支工作，但在他们使用的方法中找到了一个共同的主题。这种设置是培养博士研究人员的理想选择。他们，以及我们的博士后和顾问从相邻领域之间的相互作用极大地保护。与我们的国际合作伙伴合作，使我们的博士研究人员沉浸在其他社区，建立同行网络，并获得进一步的专业知识。申请者包括五名终身教授和四名初级科学家。按照计划，第二组有了变化。这导致了我们的研究范围的扩大。我们小组的规模足够大，因此可以组织富有成效的定期研讨会和工作组，而不会给个人带来太大的压力。我们的研究和资格认证计划使我们的博士研究人员接触到广泛的研究问题，同时传达了对我们的科学至关重要的团结感。

The research program is rooted in geometry. The research projects cover a wide range of subjects from mathematical physics to number theory, however, the methods used to study these distinct sets of problems are closely related.The most visible unifying technique used in all our projects is cohomology, a versatile tool central to all geometric disciplines, posed to gain yet more importance in years to come. Nearly all of our projects use Hodge theory, Dirac operators, deformation theory, Lie groups or algebraic geometry. This leads to synergism, of which we are taking advantage. The interplay between abstract algebra and concrete geometry is a Leitmotiv of our research.- The participating researchers work in dierent branches of geometry, yet find a common theme in the methods they use. This set-up is ideal for the training of doctoral researchers. They, as well as our postdocs and advisers prot immensely from the interactions between neighboring fields.- Collaboration with our international partners allows our doctoral researchers to immerse themselves into other communities, establish peer networks, and access further expertise.- The group of applicants consists of five tenured professors and four junior scientists. As planned, there has been a change in the second group. This has led to an expansion of our research profile.- The size of our group is big enough so that fruitful regular seminars and working groups can be organized without putting too much strain on individuals. It is small enough so that researchers know each other well.Our research and qualification program exposes our doctoral researchers to a wide range of research problems while conveying the sense of unity that is central to our science.