Tilting Theory Revisited - Developments in Real Grothendieck Groups

Explore the latest developments in tilting theory over the past decade in this 52-minute lecture by Osamu Iyama at BIMSA. Delve into the fundamental concepts of tilting complexes and their role in controlling equivalences between derived categories of rings. Examine the class of silting complexes and their relationship to tilting complexes from a mutation perspective. Focus on the significance of 2-term silting complexes and their bijective correspondence to important structures in module and derived categories. Investigate the real Grothendieck groups and their connection to 2-term silting complexes, including the concept of g-fans in finite dimensional algebras. Explore the properties of g-fans, their completeness conditions, and their extension to the whole real Grothendieck group using Asai's TF-equivalence relation. Discover the strong connection between Derksen-Fei's canonical decompositions and TF-equivalence classes, gaining insights into the representation theory of algebras and its applications in categorification of cluster algebras.