On homological dimensions and combinatorics of finite dimensional algebras

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We aim to find new connections between the homological algebra of incidence algebras of posets and their combinatorial properties. We also want to extend the theory of cluster tilting objects and higher Auslander algebras of Iyama and apply them to Cohen-Macaulay Artin algebras in the sense of Auslander and Reiten. Moreover, we want to study homological properties of the Jacobson radical and its higher syzygies that have a connection to the finitistic dimension conjecture.