Homological Mirror Symmetry and its applications

Mirror symmetry arose originally in physics, as a duality between N= 2 superconformal field theories. Witten formulated a more mathematically accessible version, in terms of topological field theories. A fundamental problem concerns the range of validity of mirror symmetry. In the world of sigma models, the traditional formulation concerns Calabi-Yau complete intersection in toric varieties, but recently far more wide-ranging constructions have been proposed. In the context of Landau-Ginzburg models, the toric picture has been described, but its limitations are far from clear. In the case where the resulting variety is Fano, several examples have been studied carefully, giving us a reasonably good idea of how mirror symmetry should work. However, one can also produce Landau- Ginzburg theories which are potentially mirrors of general type varieties, and here the picture is considerably less clear. Looking a little further, probably the most basic issue is to make the geometric picture of mirror symmetry more flexible. For instance, the effect of birational transformations on the derived categories of algebraic varieties is well-understood, but the corresponding mirror picture is just beginning to be studied. All of these questions will be studied as part of this proposal. Besides their intrinsic importance, one expects to obtain applications to classical problems in symplectic topology and algebraic geometry. Ultimately, one can imagine that mirror symmetry would become a purely mathematical tool of some importance in these fields. To be a little more explicit, we list some of the issues where we expect to make substantial progress: 1. Proving HMS in greater generality and more systematically (higher dimensional Fanos and Calabi-Yaus; better methods for constructing functors; the effect of blowups). 2. Extending the statement of HMS to varieties of general type (formulating the precise statements; conducting nontrivial tests; and proving some simple model cases). 3. Using HMS to study classical problems in algebraic geometry (in particular, to prove the nonrationality of certain algebraic varieties). 4. Developing the theory of noncommutative Hodge Structures, which should be an important tool, in particular connecting HMS with the enumerative part of mirror symmetry. 5. Clarifying the tropical Hodge conjecture. This is a three year project and it will be carried out at the Mathematical Department of University of Vienna, where have an established Algebraic Geometry group together with Professor Hauser. We plan one workshop and a bigger semester event in ESI, an application for which is being prepared by D. Auroux, L. Katzarkov, M. Kontsevich, D. Orlov. We have also established a solid connection with the String thory seminar of M. Kreuzer in TUW. Many of the activities described bellow will be joint activities with this seminar. From all we have said it becomes clear that: a) This is an international project combining efforts of the leaders in the field, representing leading research institutions, and who have a well established record of working together. b) This is a project with a clear and massive impact on several subjects - algebraic geometry , symplectic geometry, homological algebra and string theory. c) The project will have a big educational impact on several institutions in Vienna and will also enchance scientific life of the Math Community and String Theory community in the city.

Mirror symmetry arose originally in physics, as a duality between N= 2 superconformal field theories. Witten formulated a more mathematically accessible version, in terms of topological field theories. A fundamental problem concerns the range of validity of mirror symmetry. In the world of sigma models, the traditional formulation concerns Calabi-Yau complete intersection in toric varieties, but recently far more wide-ranging constructions have been proposed. In the context of Landau-Ginzburg models, the toric picture has been described, but its limitations are far from clear. In the case where the resulting variety is Fano, several examples have been studied carefully, giving us a reasonably good idea of how mirror symmetry should work. However, one can also produce Landau- Ginzburg theories which are potentially mirrors of general type varieties, and here the picture is considerably less clear. Looking a little further, probably the most basic issue is to make the geometric picture of mirror symmetry more flexible. For instance, the effect of birational transformations on the derived categories of algebraic varieties is well-understood, but the corresponding mirror picture is just beginning to be studied. All of these questions will be studied as part of this proposal. Besides their intrinsic importance, one expects to obtain applications to classical problems in symplectic topology and algebraic geometry. Ultimately, one can imagine that mirror symmetry would become a purely mathematical tool of some importance in these fields. To be a little more explicit, we list some of the issues where we expect to make substantial progress: 1. Proving HMS in greater generality and more systematically (higher dimensional Fanos and Calabi-Yaus; better methods for constructing functors; the effect of blowups). 2. Extending the statement of HMS to varieties of general type (formulating the precise statements; conducting nontrivial tests; and proving some simple model cases). 3. Using HMS to study classical problems in algebraic geometry (in particular, to prove the nonrationality of certain algebraic varieties). 4. Developing the theory of noncommutative Hodge Structures, which should be an important tool, in particular connecting HMS with the enumerative part of mirror symmetry. 5. Clarifying the tropical Hodge conjecture. This is a three year project and it will be carried out at the Mathematical Department of University of Vienna, where have an established Algebraic Geometry group together with Professor Hauser. We plan one workshop and a bigger semester event in ESI, an application for which is being prepared by D. Auroux, L. Katzarkov, M. Kontsevich, D. Orlov. We have also established a solid connection with the String thory seminar of M. Kreuzer in TUW. Many of the activities described bellow will be joint activities with this seminar. From all we have said it becomes clear that: 1. This is an international project combining efforts of the leaders in the field, representing leading research institutions, and who have a well established record of working together. 2. This is a project with a clear and massive impact on several subjects - algebraic geometry , symplectic geometry, homological algebra and string theory. 3. The project will have a big educational impact on several institutions in Vienna and will also enchance scientific life of the Math Community and String Theory community in the city.