Lagrangian Floer cohomology and Khovanov homology

Most of modern geometry studies some kind of space. The spaces considered in differential geometry are called manifolds , spaces which locally look like n-dimensional Euclidean space but globally have an interesting shape. A manifold is compact if it is closed up, with no edges. The surface of a doughnut is a compact 2-dimensional manifold. A submanifold N of a manifold M is a subset of M which is itself a manifold, usually of smaller dimension than M. There are two kinds: embedded submanifolds, which may not intersect (cross) themselves, and immersed submanifolds, which may.One usually considers manifolds with some extra geometric structure, such as a Riemannian metric , which tells you the lengths of paths in the manifold, or a symplectic structure , which tells you the areas of 2-dimensional submanifolds. Symplectic manifolds are the foundation of the mathematical formulation of mechanics, and so of much of classical physics. They are also very interesting in their own right. Mathematicians like them as they are one of very few structures with an infinite-dimensional amount of symmetry, which gives symplectic geometry an unusual, entirely global flavour. Lagrangian submanifolds are a special kind of submanifold of a symplectic manifold. Given two compact, embedded Lagrangian submanifolds L, L* of a symplectic manifold M, one can under certain conditions define the Floer cohomology groups HF(L,L*), which are roughly speaking finite-dimensional vector spaces. The definition is very difficult. To do it, one chooses an auxiliary complex structure J on M and counts J-holomorphic 2-dimensional discs D in M with boundary (edge) in the union of L and L*. The remarkable thing about HF(L,L*) is that it is independent of the choice of J, and is also unchanged by moving L and L* around amongst Lagrangian submanifolds. It encodes some mysterious, nontrivial information about Lagrangian submanifolds one cannot get at in any other known way. It is a powerful tool in symplectic geometry. In previous EPSRC-funded research, the PI and Akaho extended the definition of HF(L,L*) from embedded to immersed Lagrangians. The PI also developed new technology ( Kuranishi (co)homology ) which will simplify and streamline the definition of HF(L,L*).This proposal will exploit these ideas. We will first develop a new, simpler and more general formulation of HF(L,L*), for immersed L,L*, using the PI's new technology. Then we will apply this new formulation to four problems. The first problem will prove a conjecture about HF(L,L*) when L,L* are complex Lagrangians in a hyperkahler manifold . The point is that the new version of HF(L,L*) will have technical features which make this proof much easier than with current definitions of HF(L,L*).The second and third problems concern knot theory: the study of knots (essentially, loops of string) in 3-dimensional space. Two knots K,K* are the same if you can deform K to K* without cutting the string. It is a difficult problem to compute whether two knots are the same. Mathematicians define knot invariants , numbers etc. one can compute for a knot K, such that if the invariants of K,K* are different then K,K* are different. Two such invariants are Khovanov homology KH(K), and symplectic Khovanov homology SKH(K), which is defined by SKH(K)=HF(L,L*) for Lagrangians L,L* in a symplectic manifold M defined using K. We aim to prove the Seidel-Smith Conjecture, that KH(K)=SKH(K). This will give new insight and methods of proof in knot theory.The fourth problem uses the new version of HF(L,L*) to strengthen results of Wehrheim-Woodward relating Lagrangian Floer theory in different symplectic manifolds M_1,M_2, using Lagrangian correspondences . It shows this relation is associative , that is, going from M_1 to M_2 to M_3 is the same as going from M_1 to M_3. Here working with immersed Lagrangians is important, but current results deal only with embedded Lagrangians.

The immediate beneficiaries of this research will be Mathematicians working in Symplectic Geometry, and Mathematicians working in Knot Theory. There will be a longer term benefit to Mathematicians, and String Theorists in Theoretical Physics, interested in Mirror Symmetry of Calabi-Yau manifolds. Symplectic Geometers will benefit from the research by being in possession of a new theory of Lagrangian Floer cohomology which is more powerful, technically simpler, and easier to use than the current model (due to Fukaya, Oh, Ohta and Ono), and having some applications of this new theory to complex Lagrangians in hyperkahler manifolds, and to Lagrangian correspondences, already worked out. Knot theorists will benefit through the proof of the Seidel-Smith Conjecture, which says that a combinatorial invariant of knots, Khovanov homology, and a Floer-theoretic invariant of knots, symplectic Khovanov homology, are really the same. This means that they will be able to use techniques from Symplectic Geometry to gain new insights and prove new results about Khovanov homology which may be inaccessible using combinatorial methods. Some new results for Khovanov homology will already follow from the Seidel-Smith Conjecture and currently known results about symplectic Khovanov homology. Geometers and String Theorists interested in Mirror Symmetry will benefit because in the longer term the research will streamline and make more powerful the mathematical technology underlying Fukaya categories, which underpin the Symplectic Geometry side of the Homological Mirror Symmetry story. At present the investment of time and effort required to understand and work with Fukaya categories for non-exact symplectic manifolds is so great that very few people attempt it. By providing a simpler, shorter, less technical version of the theory, more people may enter the field. The beneficiaries will be made aware of the research by the posting of preprints on the arXiv and publication in journals, by talks by the PI, PDRA and student at seminars in UK universities and at national and international conferences, and through discussions between the PI, PDRA and their contacts in the Symplectic Geometry and Knot Theory communities. Project 1 will introduce new tools in Symplectic Geometry, which will be more powerful, cleaner, and easier to use than those versions currently available. To make it easier for people to learn and apply the new technology in Project 1, and so increase the impact of the research, the PI intends to write short introductory papers about it -- User's Guides -- as well as the long technical source papers. The PI and PDRA, and to a smaller extent the Project Student, will be undertaking the impact activities, which will primarily be writing papers, both research and expository, and giving seminar and conference talks. The PI has many years experience in these activities. He will mentor the PDRA and Project Student in academic writing, through commenting on drafts, and in giving talks, and will where possible arrange invitations for them to give seminars, for instance in the Oxford Algebraic and Symplectic Geometry seminar, co-organized by the PI, and to attend relevant conferences. The Project Student will have opportunities to speak in the Junior Geometry seminar, where graduates give lectures to each other on their research areas. In this way, the PDRA and Project Student will increase their abilities in technical writing and in giving presentations, both important transferrable skills.