Network on silting theory

• Applicant: Professor Dr. Steffen Koenig
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 451916042

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.

• DFG Programme: Scientific Networks
• Co-Investigator: Alexandra Zvonareva, Ph.D.