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)提供资金的。这个FRG建立在镜面对称性和热带几何学的各种最新成功的基础上。一方面，Strominger-Yau-Zaslow猜想导致Kontsevich、Soibelman、Gross、Siebert、Zharkov等人用积分仿射流形和热带数据来看待镜面对称性。另一方面，Mikhalkin在使用热带几何进行全纯曲线计数方面的开创性工作表明，Gromov-Witten不变量可以通过热带方法获得，并且热带方法越来越多地被视为研究代数簇的工具。这个FRG的目的是在热带几何学和镜面对称性之间建立进一步的联系，目的是创造这两个领域的新的综合。这个项目的一些研究方向包括：一般类型的Fanos和流形的镜像对称性的研究；在这种新的背景下的拉格朗日和Syz颤动；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计数几何；非对易Hodge理论的发展，热带同调及其与经典同调和Hodge猜想的关系；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，并将热带几何的应用扩展到经典的代数几何世界。该FRG成员包括：Ricardo Castano-Bernard(堪萨斯州立大学)、Mark Gross(圣地亚哥)、Ilia Itenberg(斯特拉斯堡)、Ludmil Katzarkov(迈阿密)、Viatcheslav Kharlamov(斯特拉斯堡)、Maxim Kontsevich(I.H.E.S&Amp；迈阿密)、Diego Matessi(亚历山德里亚)、Grigory Mikhalkin(日内瓦)、YanSoibelman(堪萨斯州)、Jack(希伯来大学)、Ilia Zharkov(堪萨斯州)。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建。弦理论还表明，量子理论在一定的范围内产生了“常规”几何学。然后，各种“弦对偶性”对相同的物理量给出了等价但在数学上非常不同的描述。一个美丽而深刻的例子说明了所有这些想法是镜像对称。该项目的一个重要组成部分是扩大联邦政府成员探索镜像对称热带方法的现有合作联系，使之成为一个由博士后、研究生和专家在综合研究-培训环境中组成的坚实的合作网络。这包括在地方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这一项目的关键方面之一，该项目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.