Mirror Symmetry, Hodge Theory and Differential Equations-猫眼课题宝

DOI：

10.4171/owr/2015/22

发表时间：

2015

期刊：

Oberwolfach Reports

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

C. Hertling;C. Sabbah;Christian Sevenheck

以下是Oberwolfach镜面对称、霍奇理论和微分方程研讨会（2015年4月）的报告，该研讨会汇集了来自量子上同调、复杂代数几何、霍奇理论表示理论等各个领域的研究人员。数学学科分类（2010）：14J33、32S40、14D07、34Mxx、53D45。镜像对称、霍奇理论和微分方程研讨会于2015年4月20日至24日举行，旨在报告与霍奇理论、线性微分方程、量子上同调、范畴论和表征理论等纯数学各个领域相关的研究课题的最新进展。讲习班有25名参加者，地域广阔。与会者名单中包括几位年轻的博士后，研讨会是一个极好的机会，让他们向更多的观众展示他们的成果。会议突出了这些数学领域的激烈活动，以及与其他数学主题的强烈互动，如朗兰兹对应，表示理论和不规则微分方程。讲习班包括19场演讲和许多非正式讨论，并辅以一些晚间讨论会。有些报告会概述与讲习班主题有关的某一特定领域，其他报告会报告确切的新成果，少数报告会主要载有新想法或正在进行的工作。例如，Duco van Straten谈到了与Oberwolfach报告22/2015 A的联合工作。Mellit和V. Golyshev研究几何朗兰兹对应和同余微分方程，利用量子上同调和朗兰兹对应的工具对Fano流形进行分类。cl<s:1> lia Pech和Konstanze Rietsch的两次演讲（部分是与R. Marsh和L. Williams合作的）是关于非环变的镜像对称命题，如齐次空间，更具体地说是格拉斯曼方程（包括等变方面）。一整天都在讨论各种超几何方程。Uli Walther（基于与L. Matusevich和E. Miller以及M. Schulze的联合工作）向我们介绍了Euler-Koszul同调技术，Thomas Reichelt使用这些技术来描述如何将混合Hodge模块的形式主义应用于Gelfand-Kapranov-Zelevinsky微分系统的研究。Takuro Mochizuki解释了他最近在twistor模块和gkz系统方面的工作，Alberto Castaño domdu ā nguez解释了关于Dwork族上同调的超几何描述的结果。Hiroshi Iritani报告了他最近关于涉及大量子上同环的镜像陈述的工作，而Alessandro Chiodo概述了他（与Y. Ruan和H. Iritani）关于Landau-Ginzburg/Calabi-Yau对应的结果。列夫·鲍里索夫（Lev Borisov）的谈话展示了镜像对称的一个令人惊讶的应用：从所谓的双镜像Calabi-Yau族（即具有相同镜像的族）的一些具体例子中，我们可以推导出仿射线的类是Grothendieck环中的零因子。量子上同调和镜像对称的范畴方面已经在几次演讲中进行了讨论：Dmytro Shklyarov解释了如何从范畴（如矩阵分解）（重新）构造GaußManin上同调，Etienne Mann谈到了他与M. Robalo关于Gromov-Witten不变量的范畴化的合作工作。Todor Milanov最近报道了一种基于奇异理论和Landau-Ginzburg模型中的振荡积分和周期积分的构造：它在与这些数据相关的特定Fock空间上建立了一个顶点算子代数结构。Alexey Basalaev解释了如何赋予上同场理论一个作用SL（2，C）和如何显示某些势的模块化性质。以一种不同的精神，Emmanuel Scheidegger与M. Alim， H. Movasati和st . t . Yau共同讨论了在Calabi-Yau三倍模空间上向量场的某种李代数的构造及其与全纯异常方程的关系。Ana Ros Camacho通过对两个多项式之差施加非零量子维矩阵分解的存在性，引入了一种新的齐次多项式等价关系，称为轨道等价，并考虑了ADE奇点的情况。Helge Ruddat报告了与B. Siebert关于退化的Calabi-Yau变种族正则坐标的构造的共同工作。镜像对称、霍奇理论和微分方程在一次概述演讲中，Philip Boalch描述了野生性状变异的例子，如有限维乘法辛商。他将这些例子与欧拉1764年的一篇论文联系起来，并证明欧拉的连续多项式是群值矩映射。Marco Hien在会议上的最后一次演讲是关于d 'Agnolo-Kashiwara关于任意微分系统的黎曼-希尔伯特对应的拓扑方法。总而言之，我们觉得我们的会议非常有趣，有许多精彩的演讲，涵盖了各种各样的主题。会谈期间的讨论（以及周三下午前往圣罗马的传统徒步旅行期间的讨论）非常令人兴奋。与会者的热情和MFO的良好气氛是研讨会取得成功的主要原因。这次会议表明，镜像对称的主题，在其所有分支中，一如既往地充满活力，许多悬而未决的问题仍然摆在我们面前。致谢：MFO和研讨会组织者感谢Simons基金会对望月拓郎（Takuro Mochizuki）在MFO“Simons客座教授”项目中的支持。主办方也感谢ANR-DFG项目SISYPH的财政支持（ANR-13-IS01-0001-01/02， DFG编号HE 2287/4-1和SE 1114/5-1）。1205工作坊：镜面对称，霍奇理论和微分方程

The following is the report on the Oberwolfach workshop Mirror Symmetry, Hodge Theory and Differential Equations (April 2015), which brought together researchers from various areas such as quantum cohomology, complex algebraic geometry, Hodge theory representation theory etc. Mathematics Subject Classification (2010): 14J33, 32S40, 14D07, 34Mxx, 53D45. Introduction by the Organisers The workshop Mirror Symmetry, Hodge Theory and Differential Equations, which took place from April 20 to 24, 2015 aimed at reporting on recent developments on research topics related to various areas of pure mathematics such as Hodge theory, linear differential equations, quantum cohomology, category theory and representation theory, to name only a few. The workshop had 25 participants, with a wide geographical horizon. The list of participants included several young postdocs, the workshop was an excellent opportunity for them to present their results to a larger audience. The meeting has highlighted the intense activity in these mathematical domains, as well as the strong interaction with other mathematical topics, such as Langlands correspondence, representation theory and irregular differential equations. The workshop consisted of 19 talks and many informal discussions, supplemented with some evening discussion sessions. Some talks were meant to give an overview on a particular field relevant for the main subject of the workshop, others reported on precise new results and and a few ones mainly contained new ideas or work in progress. For example, Duco van Straten talked on joint work with 1202 Oberwolfach Report 22/2015 A. Mellit and V. Golyshev on geometric Langlands correspondence and congruence differential equations, motivated by the classification of Fano manifolds by using quantum cohomology and tools from Langlands correspondence. Two talks, by Clélia Pech and Konstanze Rietsch (partly on joint work with R. Marsh and L. Williams) were on mirror symmetry statements for non-toric varieties such as homogeneous spaces and more specifically Grassmannians (including equivariant aspects). A whole day was concerned with talks related to various kinds of hypergeometric equations. Uli Walther (based on joint work with L. Matusevich and E. Miller as well as M. Schulze) introduced us to the techniques of Euler-Koszul homology, Thomas Reichelt used these to describe how the formalism of mixed Hodge modules can be applied to the study of Gelfand-Kapranov-Zelevinsky differential systems. Takuro Mochizuki explained his recent work on twistor modules and GKZ-systems, and Alberto Castaño Domı́nguez explained results on hypergeometric description of the cohomology of the Dwork family. Hiroshi Iritani reported on his recent work on mirror statements involving the big quantum cohomology rings, whereas Alessandro Chiodo overviewed his results (with Y. Ruan and H. Iritani) on the Landau-Ginzburg/Calabi-Yau correspondence. The talk of Lev Borisov showed a surprising application of mirror symmetry: From some specific examples of so-called double mirror Calabi-Yau families (i.e., families having the same mirror) one can derive that the class of the affine line is a zero divisor in the Grothendieck ring. Categorical aspects of quantum cohomology and mirror symmetry have been discussed in several talks: Dmytro Shklyarov explained how to (re)construct GaußManin cohomology from categories (like matrix factorizations), and Etienne Mann talked about his joint work with M. Robalo on categorification of Gromov-Witten invariants. Todor Milanov reported on a recent construction which builds on the oscillating integrals as well as the period integrals in singularity theory and Landau-Ginzburg models: It establishes a vertex operator algebra structure on a certain Fock space associated to these data. Alexey Basalaev explained how to endow cohomological field theories with an action of SL(2,C) and how to show modularity properties of certain potentials. In a somewhat different spirit, Emmanuel Scheidegger talked on joint work with M. Alim, H. Movasati and S.T. Yau on the construction of a certain Lie algebra of vector fields on the moduli space of Calabi-Yau threefolds and its relation to the holomorphic anomaly equations. Ana Ros Camacho introduced a new equivalence relation for homogeneous polynomials, called orbifold equivalence, by imposing the existence of a matrix factorization with nonzero quantum dimension for the difference of the two polynomials, and considers the case of ADE singularities. Helge Ruddat reported on joint work with B. Siebert on the construction of canonical coordinates for a degenerating family of Calabi-Yau varieties. Mirror Symmetry, Hodge Theory and Differential Equations 1203 In an overview talk, Philip Boalch described examples of wild character varieties as finite dimensional multiplicative symplectic quotients. He related these examples to a 1764 paper of Euler, and showed that Euler’s continuant polynomials are group valued moment maps. The last talk of the conference by Marco Hien was concerned with a topological approach to the recent work of d’Agnolo-Kashiwara on Riemann-Hilbert correspondence for arbitrary differential systems. Summarizing, we feel that we had an extremely interesting meeting with many beautiful talks covering a large variety of subjects. The discussions that took place between the talks (as well as during the traditional hike to St. Roman on Wednesday afternoon) were quite stimulating. The enthusiasm of the participants as well as the great atmosphere at MFO largely contributed to the success of the workshop. The meeting showed that the subject of mirror symmetry, in all its ramifications, is as vibrant as ever and many open questions are still ahead of us. Acknowledgements: The MFO and the workshop organizers would like to thank the Simons Foundation for supporting Takuro Mochizuki in the “Simons Visiting Professors” program at the MFO. The organizers also acknowledge the financial support of the ANR-DFG program SISYPH (ANR-13-IS01-0001-01/02, DFG No HE 2287/4-1 & SE 1114/5-1). Mirror Symmetry, Hodge Theory and Differential Equations 1205 Workshop: Mirror Symmetry, Hodge Theory and Differential Equations

Mirror Symmetry, Hodge Theory and Differential Equations

文献摘要

相似文献