Award: DMS 1108397, Principal Investigator: Garrett Alston
Mirror symmetry is an exciting branch of mathematics that is concerned with duality between the symplectic and algebraic geometry of Calabi-Yau manifolds. There are two complementary conjectures that purport to explain this duality-Kontsevich's Homological Mirror Symmetry conjecture (HMS) and the Strominger-Yau-Zaslow conjecture (SYZ). The principal investigator aims to study mirror symmetry in the context of these conjectures by studying explicit examples. First, an investigation of the manifold mirror to the quintic threefold will be undertaken. The PI will search for objects on the mirror manifold that exhibit behavior similar to a certain class of Lagrangian submanifolds on the quintic. This will provide a direct and concrete illustration of HMS. In conjunction with this work, a study of Floer cohomology of Lagrangians in K3 surfaces will be undertaken. This work will exploit explicit Lagrangian torus fibration constructions of Gross, Castano-Bernard, Matessi and others, which in turn are inspired by the SYZ conjecture.
Algebraic geometry and symplectic geometry are deep and important fields whose origins go back hundreds of years. The questions and ideas arising from these fields have inspired many mathematicians and physicists and led to great scientific advances. Mirror symmetry promises to add to this legacy. Mirror symmetry has its roots in string theory and today is an important branch of physics. To mathematicians, mirror symmetry is an exciting and tantalizing subject because it hints at a link between algebraic and symplectic geometry. It is often these types of links-links between different subjects-that lead to breakthroughs. The goal of this project is to discover links between algebraic and sympletic geometry by applying newly developed theory and techniques to certain examples that are of central importance to mirror symmetry.
Mirror symmetry aims to find hidden relationships between algebraic geometry and symplectic geometry. This project studied an important class of examples in mirror symmetry, the smoothings of A_n surfaces. These are four dimensional geometric objects. Explicitly, they are the set of solutions of certain polynomials in three variables. The main result of this project was the calculation of some important invariants associated to the A_n surfaces. More precisely, the Floer cohomology of a large class of immersed Lagrangian spheres. Floer cohomology can be thought of as an algebraic object (invariant) associated to a geometric object (Lagrangian). In general, Floer cohomology is an important invariant in symplectic geometry and mirror symmetry. Symplectic geometry has its roots in physics, classical mechanics, and dynamical systems. In this context, one application of Floer cohomology is that it can detect the existence of interesting trajectories. In terms of mirror symmetry, Floer cohomology provides the link between symplectic geometry and algebraic geometry. Floer cohomology is a very technical and difficult invariant to deal with. In order to carry out the calculation in this example, new techniques were devised that greatly simplify the difficulties involved. These new techniques are applicable to a wide range of examples beyond just those studied in the project. In fact, they are applicable to many new examples that have not received much attention yet. It is hoped that this will simplify Floer theory and make it more applicable.