Network on silting theory

淤泥理论网络

• 批准号: 451916042
• 负责人: Professor Dr. Steffen Koenig
• 金额: --
• 依托单位: Institut für Algebra und Zahlentheorie (IAZ)
• 依托单位国家: 德国
• 项目类别: Scientific Networks
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/451916042?language=en
• 关键词: Network; silting; theory

A major success story of representation theory has started with Morita theory, which decides when two algebras have the same category of representations. It continued with tilting theory, which connects two algebras having different representation categories. The derived equivalences resulting from compact tilting complexes are now the centre of a whole new area, which has found plenty of applications in all parts of representation theory and in neighbouring areas. Parallely, non-compact tilting theory has developed on its own, with connections to model theory and logic. New directions of tilting such as cluster tilting and tau-tilting are related with algebraic combinatorics and quantum algebras. Currently, a new area is emerging that extends and unifies these developments in many respects: Silting theory works with abelian and with triangulated categories, incorporates and develops fundamental structures such as torsion theories and t-structures as well as localisations and it employs higher structures. New applications are expected to other quickly emerging areas such as stability structures as well as to classical problems such as the Auslander-Reiten conjecture on stable equivalences and the Telescope conjecture.Almost all published articles in this new area have appeared since 2015, the year when in Verona a first workshop on silting theory was held, with less than fifty participants. A small workshop in China followed in 2018. In 2019 a summer school and conference 'Two weeks of silting' was held in Stuttgart, which attracted already more than a hundred participants.The network being applied for aims at strengthening the existing collaborations in this area, stimulating and supporting new collaborations and providing a basis for exchange within the area and with areas where new methods and applications are to be found.The network's objectives form three pairs of groups:- Two groups of objectives focus on the fundamentals of the theory, t-structures in triangulated categories and torsion pairs in abelian categories, and on the parallels and connections between these two concepts.- The third and the fourth group of objectives aim at strengthening and clarifying the interaction with localisation theories and at extending the whole theory by making available higher categorical structures and enhancements.- The fifth and the sixth group of objectives concentrate on applications to be obtained by building bridges with stability conditions and with stable categories and simple-minded systems.The network has been initiated and is being coordinated by young researchers. It is supported by a few experienced researchers and also by external experts. It will connect three German representation theory groups, in Bielefeld, Bonn and Stuttgart, with groups in the Czech Republic, Italy, Spain and the UK.

表征理论的一个主要成功案例是从Morita理论开始的，该理论决定了两个代数何时具有相同类别的表征。接着是倾斜理论，它连接了两个具有不同表示范畴的代数。由紧凑倾斜复合体产生的衍生等价现在是一个全新领域的中心，它在表示理论的所有部分和邻近领域中都有大量的应用。与此同时，非紧倾斜理论也有了自己的发展，它与模型理论和逻辑有联系。在代数组合学和量子代数的指导下，出现了簇倾斜和tau倾斜等新的倾斜方向。目前，一个新的领域正在出现，它在许多方面扩展和统一了这些发展：泥沙理论与阿贝尔和三角分类一起工作，结合并发展了基本结构，如扭转理论和t结构以及局部化，并采用了更高的结构。新的应用有望在其他快速出现的领域，如稳定结构，以及经典问题，如稳定等价的Auslander-Reiten猜想和望远镜猜想。几乎所有在这个新领域发表的文章都是在2015年之后发表的，那一年维罗纳举行了第一次关于泥沙理论的研讨会，参与者不到50人。随后于2018年在中国举办了一个小型研讨会。2019年，在斯图加特举办了一场名为“两周淤积”的暑期学校和会议，吸引了100多名参与者。正在应用的网络的目的是加强这一领域的现有合作，鼓励和支持新的合作，并为该领域内以及与需要寻找新方法和应用的领域进行交流提供基础。网络的目标形成了三对组：两组目标集中在理论的基础，三角分类中的t结构和阿贝尔分类中的扭转对，以及这两个概念之间的平行和联系。-第三和第四组目标旨在加强和澄清与本地化理论的相互作用，并通过提供更高的分类结构和增强来扩展整个理论。-第五和第六组目标集中于通过建立条件稳定、类别稳定和系统简单的桥梁来实现的应用。这个网络是由年轻的研究人员发起和协调的。它得到了一些经验丰富的研究人员和外部专家的支持。它将连接德国比勒费尔德、波恩和斯图加特的三个代表性理论小组，以及捷克共和国、意大利、西班牙和英国的小组。