Cluster algebras and Moduli spaces of local systems
来源：太阳成集团tyc411
发布时间：2018-08-14
1198
太阳成集团tyc411(中国)有限公司-百度百科九十周年院庆系列活动之六十八
报告人：沈临辉 博士（美国密西根州立大学）
时间：第一次：2018年8月22日下午4点-6点；第二次：2018年8月24日下午4点-6点
地点：浙江大学玉泉校区数学学院（工商楼200-9）报告厅
Title: Cluster algebras and Moduli spaces of local systems I
Abstract: Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation. An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space. This is a joint work with Alexander Goncharov.
Title: Cluster algebras and Moduli spaces of local systems II
Abstract: The Grassmannian Gr(k,n) parametrizes k-dimensional subspaces in C^n. Due to work of Scott, its homogenous coordinate ring C[Gr(k,n)] is a cluster algebra of geometric type. We introduce a periodic configuration space X(k,n) equipped with a natural potential function W. We prove that the topicalization of (X(k,n), W) canonically parametrizes a linear basis of C[Gr(k,n)], as expected by the Duality Conjecture of Fock-Goncharov. We identify the tropical set of (X(k,n), W) with the set of plane partitions. As an application, we show a cyclic sieving phenomenon involving the latter. This is joint work with Daping Weng.
Abstract: Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation. An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space. This is a joint work with Alexander Goncharov.
Title: Cluster algebras and Moduli spaces of local systems II
Abstract: The Grassmannian Gr(k,n) parametrizes k-dimensional subspaces in C^n. Due to work of Scott, its homogenous coordinate ring C[Gr(k,n)] is a cluster algebra of geometric type. We introduce a periodic configuration space X(k,n) equipped with a natural potential function W. We prove that the topicalization of (X(k,n), W) canonically parametrizes a linear basis of C[Gr(k,n)], as expected by the Duality Conjecture of Fock-Goncharov. We identify the tropical set of (X(k,n), W) with the set of plane partitions. As an application, we show a cyclic sieving phenomenon involving the latter. This is joint work with Daping Weng.