Parabolic Kostka polynomials, quiver varieties, crystal bases and tropicalcombinatorics

2003-2006年，在“抛物科斯特卡多项式、振动品种、晶体基和热带组合”项目的研究过程中，项目成员在主要数学期刊上发表论文约25篇，组织和参加了多次国际国内会议和研讨会，定期参加联合讨论和合作。2003年“抛物科斯特卡多项式、颤栗品种、晶体基和热带组合”项目的主要活动是2003年6月16日至19日在京都大学RIMS举行的“量子上同调国际研讨会”，由a.n.k riillov和m.g guest（东京城市大学）组织。研讨会汇集了该领域的几位顶尖专家，如教授。中岛h . (Kyodai)，齐藤k . (rim)，金b .（韩国），A.-L。Mare（加拿大），A.Buch（瑞典），以及许多（约50名）国内参与者。2004年8月8日至22日在京都大学RIMS举办的“热带代数几何和热带组合学”国际研讨会是“Parab…More olic Kostka多项式，振动变种，晶体基和热带组合学”项目的主要活动。研讨会汇集了多位“热带数学”领域的权威专家，如教授。A.Knutson（美国加州大学伯克利分校），E.Miller（美国明尼苏达大学），G.Mikhalkin（加拿大多伦多大学），D.Speyer（美国加州大学伯克利分校），O.Viro（瑞典乌普萨拉大学），M.Kashiwara (RIMS)， M.Okado（大阪），Y.Yamada（神户），以及众多（近60名）国内参与者。这两个研讨会都被认为是成功的，并引起了日本对热带数学和量子上同论的显著兴趣。在项目的主体部分，a.n.c riillov作为特邀演讲人参加了8月2-4日在中国南开大学举行的“组合、特殊函数和物理”国际研讨会，以及国内的一些会议。A.N.Kirillov在项目主体中获得的关于抛物型Kostka多项式的部分基本结果，已发表在《物理学报》第40卷。特别地，本文包含了所谓广义饱和猜想的一个证明，以及抛物型Kostka多项式、Schur函数等几个有趣的新性质的证明。A.N.Kirillov和T.Maeno获得并发表了关于舒伯特微积分与非交换微分之间联系的几个重要结果。特别地，我们描述了一些非交换代数变量的平面连接生成的代数，并证明了非交换B_n Schubert多项式的Monk公式。少

During 2003-2006 years in the course of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics", the members of the Project have published about 25 papers in the leading mathematical journals, have organized and attended several International and Domestic Conferences and Workshops, participated regularly in the joint discussions and collaboration.The main event of 2003 year term of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics" was the "International Workshop on Quantum Cohomology" to be held during June 16-19, 2003 at RIMS, Kyoto University, and organized by A.N.Kirillov and M.Guest (Tokyo Metropolitan University). The Workshop collected several leading specialists in the field such as Profs. H.Nakajima (Kyodai), K.Saito (RIMS), B.Kim (S.Korea), A.-L.Mare (Canada), A.Buch (Sweden), as well as many (about 50) domestic participants.The main event of 2004 year term of the Project "Parab … More olic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics" was the International Workshop "Tropical algebraic geometry and tropical combinatorics" carried out August 8-22, RIMS, Kyoto University, and organized by A.N.Kirillov and M.Noumi. The Workshop collected several leading specialist in "Tropical Mathematics" such as Profs. A.Knutson (UC Berkeley, USA), E.Miller (Univ. of Minnesota, USA), G.Mikhalkin (Toronto Univ., Canada), D.Speyer (UC Berkeley, USA), O.Viro (Uppsala Univ., Sweden), M.Kashiwara (RIMS), M.Okado (Osaka), Y.Yamada (Kobe), as well as many (near 60) domestic participants.The both Workshops were recognized to be successful and gave rise to notable interest to Tropical Mathematics and Quantum Cohomology in Japan.In the body of the Project, A.N.Kirillov attended as invited speaker the International Workshop "Combinatorics, Special Functions and Physics, 2004" held August 2-4, Nankai Univ., China, and several domestic Conferences.Part of basic results about parabolic Kostka polynomials obtained by A.N.Kirillov in the body of project, has been published in the Publications of RIMS, vol. 40. In particular, this paper contains a proof of the so-called Generalized Saturation Conjecture, as well as proofs of several new interesting properties of parabolic Kostka polynomials, Schur functions and so on.Several important results about connections between Schubert Calculus and non-commutative differential calculus have been obtained and published by A.N.Kirillov and T.Maeno. In particular, we described the algebra generated by flat connections for some noncommutative algebraic varieties, and prove Monk formula for noncommutative B_n Schubert polynomials. Less