Network on silting theory

表示理论的一个主要成功故事始于森田理论，它决定了两个代数何时具有相同的表示范畴。它继续使用倾斜理论，该理论连接了具有不同表示范畴的两个代数。由紧倾斜复形得到的等价性现在是一个全新领域的中心，它在表示理论的各个部分和邻近领域都有大量的应用。与之平行的是，非紧凑倾斜理论是独立发展起来的，与模型理论和逻辑有关。倾斜的新方向，如团簇倾斜和张倾斜，都与代数组合学和量子代数有关。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：淤积理论适用于阿贝尔和三角范畴，结合和发展了基本结构，如扭转理论和t-结构以及局部化结构，并采用了更高的结构。预计新的应用将出现在其他快速崛起的领域，如稳定结构以及经典问题，如Auslander-Reiten关于稳定等价的猜想和望远镜猜想。这一新领域几乎所有发表的文章都是从2015年开始出现的，那一年在维罗纳举行了第一次淤积理论研讨会，参与者不到50人。2018年，中国举办了一个小研讨会。2019年在斯图加特举办了一个名为“淤泥两周”的暑期班和会议，吸引了100多人参加。申请的网络旨在加强这一领域的现有合作，刺激和支持新的合作，并为该地区内以及与发现新方法和新应用的领域提供交流的基础。该网络的目标形成了三对小组：-两组目标集中在理论基础上，三角范畴中的t-结构和阿贝尔范畴中的扭转对，-第三和第四组目标旨在加强和澄清与本地化理论的互动，并通过提供更高的范畴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具有稳定条件、稳定类别和简单系统的桥梁来获得应用。该网络已经启动，并正在由年轻的研究人员协调。它得到了一些有经验的研究人员的支持，也得到了外部专家的支持。它将把比勒费尔德、波恩和斯图加特的三个德国代表理论团体与捷克共和国、意大利、西班牙和英国的团体联系起来。

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.