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Positive divisors in symplectic geometry

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Science China Mathematics 2013年第6期56卷 1129-1144页

作者： HU JianXun ruan yongbin Department of Mathematics Sun Yat-sen University Department of Mathematics University of MichiganAnn Arbor MI 48109-1109 USA

In this paper, we first prove a vanishing theorem of relative Gromov-Witten invariant of Pl-bundle. Based on this vanishing theorem and degeneration formula, we obtain a comparison theorem between absolute and relativ... 详细信息

关键词： Gromov-Witten invariant degeneration formula blow-up

The Moduli Space in the Gauged Linear Sigma Model

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Chinese Annals of Mathematics,Series B 2017年第4期38卷 913-936页

作者： Huijun FAN Tyler JARVIS yongbin ruan School of Mathematical Sciences Peking University Beijing 100871 China. Department of Mathematics Brigham Young University Provo UT 84602 USA. Department of Mathematics University of Michigan Ann Arbor MI 48105 USA.

This is a survey article for the mathematical theory of Witten's Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper the authors... 详细信息

关键词： Landau-Ginzburg Calabi-Yau Gromov-Witten GLSM Moduli Phase Variation of geometric invariant theory Symplectic reduction

Witten's D_4 Integrable Hierarchies Conjecture

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Chinese Annals of Mathematics,Series B 2016年第2期37卷 175-192页

作者： Huijun FAN Amanda FRANCIS Tyler JARVIS Evan MERRELL yongbin ruan School of Mathematical Sciences Peking University Department of Mathematics Brigham Young University Department of Mathematics University of Michigan Ann Arbor the Yangtz Center of Mathematics at Sichuan University

The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-ruan-Witten for D4 with symmetry group J and for D4T with symmetry group Gmax, respectively, are both tau-functions of the D4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in the article by Fan-Jarvis-ruan（2013）, of the Witten Integrable Hierarchies Conjecture for all simple（ADE） singularities.

关键词： Quantum cohomology Frobenius manifolds Singularity theory Integrable hierarchies

Uniruled Symplectic Divisors

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Communications in Mathematics and Statistics 2013年第2期1卷 163-212页

作者： Tian-Jun Li yongbin ruan School of Mathematics University of MinnesotaMinneapolisMN 55455USA Department of Mathematics University of MichiganAnn ArborMIUSA

In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational *** particular,we generalizeMaulik-Pandharipande’s relative/absolute correspondence to relative-divisor/absolute ***,we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.

关键词： Birational symplectic geometry Gromov–Witten invariants Symplectic divisor Uniruled invariant

Verlinde/Grassmannian Correspondence and Rank 2 δ-Wall-Crossing

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Peking Mathematical Journal 2023年第1期6卷 217-306页

作者： yongbin ruan Ming Zhang Institute for Advanced Study in Mathematics Zhejiang UniversityHangzhou 310058China Department of Mathematics University of CaliforniaSan DiegoJollaCA 92093USA

Motivated by Witten’s work,we propose a K-theoretic Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the *** recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated with the *** construct two families of stability conditions connecting the two theories and prove two wall-crossing *** confirm the Verlinde/Grassmannian correspondence in the rank two case.

关键词： Verlinde/Grassmannian correspondence Quantum K-theory Gauged linear sigma model

Mirror symmetry for special nilpotent orbit closures In Memory of Professor Gang Xiao

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Science China Mathematics 2024年

作者： Baohua Fu yongbin ruan Yaoxiong Wen AMSS HLM and MCM Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Institute for Advanced Study in Mathematics Zhejiang University Korea Institute for Advanced Study

Motivated by geometric Langlands, we initiate a program to study the mirror symmetry between nilpotent orbit closures of a semisimple Lie algebra and those of its Langlands dual. The most interesting case is Bn via Cn. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special ***, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. In the process, we reveal some very interesting and yet subtle structures of these finite covers, which are related to Lusztig’s canonical quotients of special nilpotent orbits. For example, there is a mysterious asymmetry in the footprint or range of degrees of these finite covers. Finally, we provide two examples to show that the mirror symmetry fails outside the footprint.

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Preface of the Special Issue for Banghe Li's 80th Anniversary

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Acta Mathematica Sinica,English Series 2022年第10期38卷 1665-1665页

作者： Fuquan Fang yongbin ruan Jie Wu Liqun Zhang Capital Normal University Zhejiang University Hebei Normal University Chinese Academy of Sciences

This special issue of Acta Mat hematic a Sinica,English Series dedicated to Banghe Li for his 80th birthday in *** Li was born on 7 July 1942,in Yueqing City,Zhejiang Province,*** graduated from the University of Science and Technology of China in 1965,and worked at the Chinese Academy of Sciences since *** was elected to be an academician of the Chinese Academy of Sciences in *** Banghe Li devoted great efforts to the unity of mathematics,with making his contributions in several areas of mathematics,particularly in low dimensional topology and differential *** published two books and over one hundred research articles.

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Gerbes and twisted orbifold quantum cohomology

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Science China Mathematics 2008年第6期51卷 995-1016页

作者： PAN JianZhong~1 ruan yongbin~(2,3) YIN XiaoQin~(1,3+) 1 Institute of Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,China 2 Department of Mathematics,University of Wisconsin-Madison,WI 53706,USA 3 Department of Mathematics,The Hong Kong University of Science and Technology,Hong Kong,China Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China Department of Mathematics University of Wisconsin-Madison WI 53706 Madison USA Department of Mathematics The Hong Kong University of Science and Technology Hong Kong China

In this paper,we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.

关键词： gerbe orbifold quantum cohomology 53D45

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辛双有理几何的若干研究进展

中国科学：数学 2024年第12期54卷 2099-2120页

作者：胡建勋李天军阮勇斌章唯一中山大学数学学院广州510275 School of Mathematics University of MinnesotaMinneapolisMN 55455USA 浙江大学数学高等研究院杭州310058 Mathematics Institute University of WarwickCoventryCV47ALUK

20世纪80年代,日本数学家Mori提出了极小模型纲领,其核心思想之一是把代数簇二分为单直纹簇和非单直纹簇来研究.对单直纹簇来说,试图理解以Fano簇为纤维的纤维化结构;而对于非单直纹簇,则要寻找它的极小模型.辛双有理几何是研究辛流形... 详细信息

20世纪80年代,日本数学家Mori提出了极小模型纲领,其核心思想之一是把代数簇二分为单直纹簇和非单直纹簇来研究.对单直纹簇来说,试图理解以Fano簇为纤维的纤维化结构;而对于非单直纹簇,则要寻找它的极小模型.辛双有理几何是研究辛流形的双有理等价分类,并尝试将Mori的分类理论拓展到辛几何领域.本文综述该领域的最新进展,内容包括双有理配边等价、辛单直纹和有理连通辛流形、4维辛流形的Kodaira维数及近复流形的双有理几何等.

关键词：双有理配边单直纹辛流形有理连通辛流形 Gromov-Witten不变量 Kodaira维数近复流形