Casals #2: Applications of the relation between cluster algebras and symplectic topology
Title Casals #2: Applications of the relation between cluster algebras and symplectic topology Abstract Casals #2: In this third talk, we will explore some consequences of the fact that these moduli of Lagrangian fillings have coordinate rings that are cluster algebras. Applications to symplectic topology include the detection of infinitely many Lagrangian fillings (and closed Lagrangian surfaces in simple Weinstein 4-folds) and the existence of holomorphic symplectic structures on these moduli. Applications to cluster algebras, using symplectic topology, include the construction of cluster algebra structures on the coordinate ring of any Richardson variety, the existence of Donaldson-Thomas transformations, and a geometric source of examples that naturally explain quasi-cluster structures. Questions 03:13:26 Yoon Jae Nho: why do you expect only finitely many fillings for these links? 03:49:23 Jae Hee Lee: I have a naive question from part 2: can our holomorphic symplectic manifold be equipped with even more structure (such as a hyperKahler metric, or a holomorphic Lagrangian fibration...?) 03:53:33 Yoon Jae Nho: when do we know a given exact Lag filling comes from a weave? 03:54:51 Jae Hee Lee: From part 3, may I ask what the braid looks like for the dual cluster variety?
Title Casals #2: Applications of the relation between cluster algebras and symplectic topology Abstract Casals #2: In this third talk, we will explore some consequences of the fact that these moduli of Lagrangian fillings have coordinate rings that are cluster algebras. Applications to symplectic topology include the detection of infinitely many Lagrangian fillings (and closed Lagrangian surfaces in simple Weinstein 4-folds) and the existence of holomorphic symplectic structures on these moduli. Applications to cluster algebras, using symplectic topology, include the construction of cluster algebra structures on the coordinate ring of any Richardson variety, the existence of Donaldson-Thomas transformations, and a geometric source of examples that naturally explain quasi-cluster structures. Questions 03:13:26 Yoon Jae Nho: why do you expect only finitely many fillings for these links? 03:49:23 Jae Hee Lee: I have a naive question from part 2: can our holomorphic symplectic manifold be equipped with even more structure (such as a hyperKahler metric, or a holomorphic Lagrangian fibration...?) 03:53:33 Yoon Jae Nho: when do we know a given exact Lag filling comes from a weave? 03:54:51 Jae Hee Lee: From part 3, may I ask what the braid looks like for the dual cluster variety?