报告题目: Study on cluster algebras via Newton polytopes:

Laurent recurrence formula, positivity and polytope basis (joint work with Jie Pan)

报告人：李方教授（浙江大学）

报告时间：12月9日14:30-15:20 腾讯会议：744 152 454

报告摘要:In this paper, we study the Newton polytopes of F-polynomials in totally

sign-skew-symmetric cluster algebras and generalize them to a larger set consisting of polytopes N_h associated to vectors h\in Z^n as well as S consisting of polytope functions \rho_h corresponding to N_h. The main contribution contains that

(i) obtaining a recurrence construction of the Laurent expression of a cluster variable in a cluster from its g-vector;

(ii) proving the subset P of S is a strongly positive basis of U(A) consisting of certain indecomposable universally positive elements, which is called as the polytope basis;

(iii) constructing some explicit maps among corresponding F-polynomials, g-vectors and d-vectors to characterize their relationship.

  As an application of (i), we give a affirmation for the positivity conjecture of cluster variables in a totally sign-skew-symmetric cluster algebra, which in particular provides a new method  different with that given in [GHKK] to present the positivity of cluster variables in the skew-symmetrizable case. As another application, a conjecture on Newton polytopes posed by Fei is answered affirmatively.  In rank 2 case, S coincides with the greedy basis introduced by Lee, Li and Zelevinsky. Hence, we can regard S as a natural generalization of the greedy basis in general rank.

报告人简介：李方，浙江大学教授、博士生导师，现任浙大高等数学研究所所长、浙江大学代数与几何基层教学组织责任教授，代数课程教学改革负责人。入选教育部“新世纪优秀人才支持计划"和省151人才工程人选。研究领域先后涉及代数与表示论多个方面，近年主要研究丛代数理论，解决了一系列重要的公开问题，形成了自己的研究特色，推动了丛代数领域的发展。培养出了一些优秀的、有较大学术影响的年轻人。已先后主持国家自然科学基金七项和浙江省自然科学基金重大和重点项目各一项。曾获浙江省高校科技进步一等奖等奖项。任中国高等教育学会教育数学专委会常务副理事长、中国数学会理事。中国数学会名词审定委员会委员。至今共发表论文130余篇，其中90余篇在SCI刊物上，包括Adv. Math., Trans. AMS, Comm. Math. Phys., Compositio Math., Math. Annalen 等有重要影响的顶级期刊。任Electronic Research Archive(SCI)、数学与人文丛书、浙江大学学报（理学版）编委。