Lagrangian Floer cohomology and Khovanov homology

大多数现代几何学研究某种空间。微分几何中考虑的空间称为流形，局部看起来像n维欧氏空间，但整体上有一个有趣的形状。一个流形是紧致的，如果它是封闭的，没有边。甜甜圈的表面是一个紧凑的二维流形。一个流形M的一个子流形N是M的一个子集，它本身也是一个流形，通常维数比M小。有两种：嵌入子流形（embedded submanifold），可能不相交（cross）;浸入子流形（immersed submanifold），可能相交。人们通常考虑具有一些额外几何结构的流形，例如黎曼度量，它告诉你流形中路径的长度，或者辛结构，它告诉你二维子流形的面积。辛流形是力学数学公式的基础，也是许多经典物理学的基础。它们本身也很有趣。数学家们喜欢它们，因为它们是极少数具有无限维对称性的结构之一，这给辛几何带来了一种不寻常的、完全全局的味道。拉格朗日子流形是辛流形的一种特殊子流形。给定辛流形M的两个紧致嵌入拉格朗日子流形L，L*，在一定条件下可以定义Floer上同调群HF（L，L*），它们大致上是有限维向量空间。这个定义很难。为了做到这一点，我们在M上选择一个辅助复结构J，并计算M中的J-全纯二维圆盘D，其边界（边）在L和L* 的并集中。HF（L，L*）的显著之处在于它与J的选择无关，并且在拉格朗日子流形之间移动L和L* 也不会改变。它编码了一些关于拉格朗日子流形的神秘的、非平凡的信息，人们无法以任何其他已知的方式获得这些信息。它是辛几何中的一个强有力的工具。在以前EPSRC资助的研究中，PI和Akaho将HF（L，L*）的定义从嵌入拉格朗日扩展到浸入拉格朗日。PI还开发了新技术（Kuranishi（co）homology），该技术将简化和简化HF（L，L*）的定义。我们将首先开发一个新的，更简单和更普遍的公式HF（L，L*），浸没L，L*，使用PI的新技术。然后，我们将把这个新公式应用于四个问题。第一个问题是证明关于HF（L，L*）的一个猜想，当L，L* 是超kahler流形上的复拉格朗日量时.问题的关键是，新版本的HF（L，L*）将有技术特点，使这个证明比HF（L，L*）的现有定义容易得多。第二和第三个问题涉及纽结理论：研究三维空间中的纽结（本质上是弦的环）。两个结K，K* 是相同的，如果你可以变形K到K*，而不切断字符串。计算两个节点是否相同是一个困难的问题。数学家定义了结不变量、数等，可以计算结K，使得如果K、K* 的不变量不同，则K、K* 也不同。两个这样的不变量是Khovanov同调KH（K）和辛Khovanov同调SKH（K），其定义为对于在使用K定义的辛流形M中的拉格朗日L，L* 的SKH（K）=HF（L，L*）。我们的目标是证明Seidel-Smith猜想，KH（K）=SKH（K）。第四个问题利用新的HF（L，L*）形式，利用Lagrange对应，在不同的辛流形M_1，M_2上加强了Wehrheim-Woodward关于Lagrange Floer理论的结果。它表明这种关系是结合的，即从M_1到M_2到M_3与从M_1到M_3是相同的。在这里工作与沉浸拉格朗日是重要的，但目前的结果只处理嵌入拉格朗日。

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.