FRG: Collaborative Research: Mirror Symmetry & Tropical Geometry

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这个FRG建立在最近在镜像对称和热带几何方面的各种成功的基础上。一方面，斯特罗明格-你-扎斯洛猜想促使Kontsevich、Soibelman、Gross、Siebert、Zharkov等人从整体仿射流形及其上的热带数据的角度来观察镜像对称性。另一方面，米哈尔金在使用热带几何的全纯曲线计数方面的开创性工作表明，通过热带方法可以获得Gromov-Witten不变量，并且越来越多地，热带方法被视为研究代数变体的工具。这个FRG的目的是在热带几何和镜像对称之间建立进一步的联系，目的是创造这两个领域的新综合。本课题的研究方向包括：范诺和一般型流形的镜像对称性研究；拉格朗日和SYZ颤振；超kaehler流形中的非阿基米德可积系统实曲线和复曲线的热带枚举几何非交换霍奇理论的发展，热带同调及其与经典同调的关系，霍奇猜想；Welschinger不变量，开放Gromov-Witten理论及其在镜像对称中的应用。这将极大地扩展镜像对称的领域和热带几何的应用到经典的代数几何世界。该小组成员包括：Ricardo Castano-Bernard（堪萨斯州）、Mark Gross（圣地亚哥）、Ilia Itenberg（斯特拉斯堡）、ludmilkatzarkov（迈阿密）、Viatcheslav Kharlamov（斯特拉斯堡）；Maxim Kontsevich （i.h.e.s. &; Miami）、Diego Matessi（亚历山德里亚）；格里戈里·米哈尔金（日内瓦），Yan Soibelman（堪萨斯州立大学），Jake Solomon（希伯来大学），Ilia Zharkov（堪萨斯州立大学）。在过去的25年里，弦理论和几何之间的密切互动导致了一个全新的数学领域的创造。弦理论还表明，“传统”几何在一定限度内是从量子理论中产生的。然后，各种“弦对偶性”给出了相同物理量的等价但在数学上非常不同的描述。一个美丽而深刻的例子就是镜像对称。该项目的一个重要组成部分是将探索热带镜像对称方法的FRG成员现有的合作联系扩展为博士后、研究生和专家在综合研究培训环境中的坚实合作网络。这包括在当地联邦政府节点组织讲习班、暑期学校和当地系列研讨会。青年和高级研究人员的国际交流是该项目的关键方面之一，将欧洲和美国的知名团体聚集在一起。将促进来自代表性不足群体的数学家和年轻研究人员的参与。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This FRG builds on various recent successes in mirror symmetry and tropical geometry. On one hand, the Strominger-Yau-Zaslow conjecture has led to work by Kontsevich, Soibelman, Gross, Siebert, Zharkov and others to view mirror symmetry in terms of integral affine manifolds and tropical data on them. On the other hand, Mikhalkin's pioneering work on holomorphic curve counting using tropical geometry demonstrated that Gromov-Witten invariants were accessible by tropical methods, and increasingly, tropical methods are being seen as a tool for studying algebraic varieties. The aim of this FRG is to make further connections between tropical geometry and mirror symmetry with the aim of creating a new synthesis of these two fields. Some of the research directions of this project include the study of mirror symmetry for Fanos and manifolds of general type; Lagrangians and SYZ fibrations in this new setting; non-archimedean integrable systems in hyper-Kaehler manifolds; tropical enumerative geometry of real and complex curves; the development of non-commutative Hodge theory, tropical homology and its relation to classical homology and the Hodge conjecture; Welschinger invariants, open Gromov-Witten theory and their applications to mirror symmetry. These should greatly extend the realm of mirror symmetry and applications of tropical geometry to the classical algebro-geometric world. Members of this FRG include: Ricardo Castano-Bernard (Kansas State), Mark Gross (San Diego), Ilia Itenberg (Strasbourg), Ludmil Katzarkov (Miami), Viatcheslav Kharlamov (Strasbourg); Maxim Kontsevich (I.H.E.S & Miami), Diego Matessi (Alessandria); Grigory Mikhalkin (Geneva), Yan Soibelman (Kansas State), Jake Solomon (Hebrew U), Ilia Zharkov (Kansas State).During past 25 years there has been intensive interaction between string theory and geometry which has led to a creation of entirely new mathematical areas. String theory also suggested that "conventional"geometry emerges from the quantum theory at certain limits. Then various "string dualities" give equivalent but mathematically very different descriptions of the same physical quantities. A beautiful and deep example illustrating all these ideas is mirror symmetry. An important component of this project is to expand the existing collaborative links of the FRG members exploring tropical methods of mirror symmetry into a solid collaborative network of postdocs, graduate students and experts in an integrated research-training environment. This includes the organization of workshops, summer schools and local seminar series at the local FRG nodes. International exchange of young and senior researchers is one of the key aspects of this project bringing together well-established groups in Europe and in the U.S. The participation of mathematicians and young researchers from underrepresented groups will be promoted.