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Cluster algebras and Moduli spaces of local systems

编辑:xyx 时间:2018年08月14日 访问次数:485

浙江大学数学科学学院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.