String Topology, J-holomorphic Curves, and Symplectic Geometry

微分几何研究的是被称为“流形”的多维弯曲空间。球体或环面（甜甜圈）的表面是二维流形的例子。通常会在流形上加上额外的几何结构。辛流形是具有“辛结构”的流形。它们是力学（物体运动的基本物理）和量子力学的数学基础的一部分。如果M是辛流形，则M中的“拉格朗日子流形”N是M中的流形N，具有M的一半维数，以某种方式与辛结构相容。它们是辛几何中的中心物体。拓扑学研究包括流形在内的空间的“形状”，关注通过拉伸或弯曲空间而不变的定性性质。二维流形的“孔数”（属）是一个拓扑不变量。球面没有孔，环面只有一个孔，所以它们作为拓扑空间是不同的。空间M的一个重要拓扑不变量是它的“同调”H(M)。H(M)是19世纪由亨利·庞加莱（Henri Poincare）引入的一个代数对象，它测量M中的“孔数”之类的东西。如果M是一个空间，那么“环路空间”LM就是M中所有环路（圆）的空间。例如，如果M是我们生活的三维空间，那么（嵌入的）环路就是结。环空间在拓扑和物理学中的弦理论中都很重要，弦理论不是将基本粒子建模为点，而是将其建模为微小的“弦环”。1999年，Chas和Sullivan发现，如果M是流形，那么环路空间LM的同调H（LM）携带了一些额外的代数结构，这些结构来自于M中环路族的相互交点——基本上，H(M)上有一个乘法。这是出乎意料和令人兴奋的，并已发展成为一个主题，称为“弦拓扑”。同调H(M)是用更基本的对象C(M)定义的，称为“链配合物”。有许多不同的方法来定义链式复合体C(M)，但它们都计算相同的同源性H(M)。弦拓扑的一个难点是，到目前为止，还没有很好的方法来定义计算H（LM）的链复合体C（LM）上的弦拓扑乘法，它只能直接定义在同构H（LM）上。弦拓扑学家认为，如果可以在链C（LM）的层次上定义操作，那么强拓扑中更深层次的代数结构将被揭示，从而从LM中定义拓扑共形场论（TCFT，一种来自量子物理的数学对象）。我们的第一个项目是定义一个计算H（LM）的链复合体C(LM)，在此基础上我们可以定义链上的字符串拓扑乘法的一个版本，并因此构造（至少是属0部分）TCFT。我们将使用PI最近定义的一种新的同调理论，称为Kuranishi同调。Kuranishi同调具有这样的性质：与“横向性”有关的问题通常会消失。由于定义链上的弦拓扑操作的困难主要是关于横向性，Kuranishi同调是一个很好的选择。现在设M是辛流形，N是拉格朗日子流形。辛几何中的一个主要工具是研究“j -全纯曲线”，即M中的二维流形C，其边界dC是n中的一维流形（环路），因此存在一个从j -全纯曲线族到环路空间LN的自然映射，将C映射到dC。我们的第二个项目是证明深谷的一些猜想。这些表明，使用（M，N）中的j全纯曲线族，我们可以定义C（LN）中的链，满足涉及第一个项目中定义的链级弦拓扑操作的方程。通过结合LN拓扑的事实，我们期望证明简单辛流形（如平面空间和射影空间）中拉格朗日子流形拓扑的重要新限制。

Differential geometry is the study of multi-dimensional curved spaces called "manifolds". The surfaces of a sphere or a torus (doughnut) are examples of 2-dimensional manifolds. Often one puts extra geometric structures on manifolds. "Symplectic manifolds" are manifolds with a "symplectic structure". They are part of the mathematical foundations of mechanics -- the basic physics of how objects move -- and quantum mechanics. If M is a symplectic manifold, a "Lagrangian submanifold" N in M is a manifold N inside M, with half the dimension of M, compatible with the symplectic structure in a certain way. They are central objects in symplectic geometry. Topology studies the "shape" of spaces including manifolds, focussing on qualitative properties that are unchanged by stretching or bending the space. The "number of holes" (genus) of a 2-dimensional manifold is a topological invariant. The sphere has no holes, and the torus has one hole, so they are different as topological spaces.One important topological invariant of a space M is its "homology" H(M). Introduced by Henri Poincare in the 19th century, H(M) is an algebraic object which measures things like the "number of holes" in M.If M is a space, the "loop space" LM is the space of all loops (circles) inside M. For example, if M is the 3-dimensional space we live in, then (embedded) loops are knots. Loop spaces are important in topology, and also String Theory in Physics, which models elementary particles not as points but as tiny "loops of string".In 1999, Chas and Sullivan discovered that if M is a manifold, then the homology H(LM) of the loop space LM carries some extra algebraic structures coming from intersections of families of loops in M with each other -- basically, H(M) has a multiplication on it. This was unexpected and exciting, and has grown into a subject called "String Topology".The homology H(M) is defined using a more basic object C(M) called a "chain complex". There are many different ways to define a chain complex C(M), but they all compute the same homology H(M). One difficulty in String Topology is that, up to now, there is no nice way to define the String Topology multiplication on a chain complex C(LM) computing H(LM) -- it is only defined directly on homology H(LM). String topologists believe that if one could define the operations on the level of chains C(LM), then deeper algebraic structures in Strong Topology would be revealed, leading to the definition of a Topological Conformal Field Theory (TCFT, a mathematical object coming out of quantum physics) from LM.Our first project is to define a chain complex C(LM) which computes H(LM), upon which we can define a version of the String Topology multiplication on chains, and so construct (at least the genus 0 part of) the TCFT.We will do this using a new homology theory called "Kuranishi homology" recently defined by the PI. Kuranishi homology has the property that problems to do with "transversality" usually disappear. Since the difficulty of defining string topology operations on chains is mostly about transversality, Kuranishi homology is a good choice.Now let M be a symplectic manifold, and N a Lagrangian submanifold. A major tool in symplectic geometry is the study of "J-holomorphic curves", 2-dimensional manifolds C in M whose boundary dC is a 1-dimensional manifold (loop) in N. So there is a natural map from families of J-holomorphic curves to the loop space LN, mapping C to dC.Our second project is to prove some conjectures by Fukaya. These say that using the families of J-holomorphic curves in (M,N), we can define chains in C(LN) satisfying equations involving the chain-level String Topology operations defined in the first project. By combining this with facts about the topology of LN, we expect to prove important new restrictions on the topology of Lagrangian submanifolds in simple symplectic manifolds, such as flat spaces and projective spaces.