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.
