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일정시작 2020-09-25
배경색상 #036eb7

Date: 9월25일 금요일 12일 오후2시
Place: 27동 220호

Speaker: 김명호

Title: Introduction to cluster algebras (and monoidal categorification)



Cluster algebra는 Q(x_1,x_2,...,x_m)  (변수가 m개인  유리함수들의 ring)의 subring으로서 "cluster variable"이라고 불리는 특별한 원소들이 생성하는 것으로 정의합니다. 이 때 변수들의 집합 {x_1,x_2,...x_m}은 "initial cluster", 그리고 x_i 들은 "initial cluster variable"이라고 각각 불립니다.  Cluster variable은 이 initial cluster에서 시작하여, 한번에 하나의 variable을 다른 것으로 바꾸는 "mutation"이라는 특별한 과정으로 얻어지는 원소를 말합니다.  따라서 이 mutation이 무엇인지 이해하면 cluster algebra의 정의를 이해할 수 있습니다. 편의상 Fomin-Zeleveinsky의 분류에서는 "skew-symmetric cluster algebra of geometric type"에 해당하는 경우만을 다루도록 하겠습니다. 이 경우에는 mutation이 "quiver mutation"이라는 조합적인 방법으로 주어집니다. Finite type classification, Laurent phenomenon등을 설명드리고, 그리고 나서도 시간이 있다면, cluster algebra의 monoidal categorification을 간략히 소개하겠습니다.



Date: 9월25일 금요일 12일 오후4시
Speaker: 조윤형

Title: Cluster structures in mirror symmetry


In this talk, I will explain "mutations" on various objects; mutations of (1) Laurent polynomials, (2) polytopes, (3) Lagrangian tori

in symplectic manifolds. I will also explain how those concepts are "expected" to fit together in the context of SYZ mirror symmetry. 

If time permits, I will provide examples of flag varieties in which every mutation appeared are turned out to coincide with the mutations in cluster theory.


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