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FRG建立在镜像对称和热带几何的各种最近的成功。一方面，Strominger-Yau-Zaslow猜想导致了Kontsevich、Soibelman、Gross、Siebert、Zharkov等人的工作，他们用积分仿射流形和热带数据来观察镜像对称。另一方面，Mikhalkin的开创性工作全纯曲线计数使用热带几何表明，Gromov-Witten不变量是可访问的热带方法，并越来越多地，热带方法被视为一种工具，研究代数簇。这个联邦德国的目的是使热带几何和镜像对称之间的进一步联系，目的是创造一个新的综合这两个领域。该项目的一些研究方向包括Fanos和一般类型流形的镜像对称性研究;在这种新的设置下的拉格朗日和SYZ纤维化;超Kaehler流形中的非阿基米德可积系统;真实的和复曲线的热带计数几何;非交换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性中的应用。这将极大地扩展镜像对称的领域和热带几何在经典代数几何世界中的应用。该联邦政府的成员包括：里卡多·卡斯塔诺-伯纳德（堪萨斯州）、马克·格罗斯（圣地亚哥）、伊利亚伊滕贝格（斯特拉斯堡）、卢德米尔·卡察尔科夫（迈阿密）、维亚切斯拉夫·哈拉莫夫（斯特拉斯堡）、马克西姆·孔采维奇（I.H.E. S迈阿密）、迭戈·马&泰西（亚历山德里亚）; Grigory Mikhalkin（日内瓦），Yan Soibelman（堪萨斯州），杰克所罗门（希伯来文U），伊利亚·扎尔科夫（堪萨斯州）。在过去的25年里，弦理论和几何学之间有着密切的相互作用，这导致了一个全新的数学领域的创建。弦理论还指出，量子理论在某些极限下会产生“常规“几何。然后，不同的“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。镜像对称是一个美丽而深刻的例子，可以说明所有这些观点。该项目的一个重要组成部分是扩大联邦德国成员的现有合作联系，探索镜像对称的热带方法，成为一个坚实的合作网络的博士后，研究生和专家在一个综合的研究培训环境。这包括在联邦共和国地方节点组织讲习班、暑期学校和地方研讨会系列。年轻和高级研究人员的国际交流是该项目的关键方面之一，汇集了欧洲和美国的知名团体，将促进来自代表性不足的团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.