Analysis of J-holomorphic Curves

AbstractProposal：DMS-9803554首席研究员：托马斯帕克这个项目涉及伪全纯曲线理论的分析方面。 目的是发展计算辛流形的Gromov-Witten不变量和代数流形的可数不变量的有效方法。 本研究的主要目标是继续开展E. Ionel关于Gromovin变式在“辛联络和”运算下的胶合公式. 其思想是将连通和收缩到一个奇异流形上，同时保持全纯曲线沿着的轨迹。 帕克和Ionel已经得到了一个特殊情况下的胶合公式，这就意味着著名的Caporaso-Harris最近公式。 一般的胶合公式应该是计算Gromov和可数不变量的一个有效的新工具。 该项目的第二部分试图回答以下问题。 设C是对某个非一般J是J-全纯的曲线，若将J扰动到一个邻近的一般J '上，有多少条J'-全纯曲线靠近C？枚举几何中的几个著名问题，有些已解决，有些未解决，都归结为这个问题。 帕克和艾诺尔有一种基于“陶伯斯障碍束”的方法。 该项目的最后一部分建议使用修改的Gromov不变量来获得只存在于特殊类的几乎复杂的结构中的曲线的不变量。数学中最基本的问题之一是确定多项式方程组的解，而实现这一目标的重要的第一步是确定解的个数。 本文给出了一组n元多项式同时解的个数的显式公式。 然后可以求n-1个变量的n个多项式的解的个数。 在这种情况下，有一个自由参数，所以解的轨迹将是曲线的并集。 有几个？ 这个问题已经被系统地研究了100多年，但只有少数特殊情况得到了解决。 然后，在1990年左右，人们意识到这些问题可以转化为辛几何，然后使用数学规范理论的强大机器来解决。 （规范理论最初是物理学的一部分，在过去的20年里，它一直是数学家和物理学家之间许多富有成果的互动的焦点;它包括杨-米尔斯和塞伯格-威滕理论，以及弦论）。 这种“格罗莫夫不变量”的方法很快导致公式回答了一些原始的枚举问题，有明确的迹象表明，还有更多的发现。 本项目的目的是进一步发展辛规范理论，以产生更多的一般公式，并将这些公式融合到相干理论中。

AbstractProposal: DMS-9803554Principal Investigator: Thomas ParkerThis project involves analytic aspects of the theory of pseudo-holomorphic curves. The aim is to develop effective methods forcomputing Gromov-Witten invariants of symplectic manifolds andenumerative invariants of algebraic manifolds. The main thrust is acontinuing project with E. Ionel on a gluing formula for Gromovinvariants under the operation of `symplectic connect sum'. The ideais to collapse the connect sum to a singular manifold, keeping trackof the holomorphic curves along the way. Parker and Ionel havealready obtained a gluing formula in special cases; these imply thewell-known recent formula of Caporaso-Harris. A general gluingformula should be an effective new tool for computing Gromov andenumerative invariants. A second part of the project seeks to foranswer the following question. Suppose C is curve that isJ-holomorphic for some non-generic J. If one perturbs J to a nearbygeneric J', how many J'-holomorphic curves are there close to C?Several well-known problems in enumerative geometry, some solved, someunsolved, reduce to this problem. Parker and Ionel have an approachbased on the `Taubes obstruction bundle' . The last part of theproject suggests using modified Gromov invariants to obtain invariantsthat count curves which exist only for special classes of almostcomplex structures.One of the most basic problems in mathematics is to determine thesolutions of a system of polynomial equations, and an important firststep toward that goal is to determine the NUMBER of solutions. Thereis an explicit formula for the number of simultaneous solutions of aset of n polynomials in n variables. One can then ask for the numberof solutions for n polynomials in n-1 variables. In this case thereis a free parameter, so the locus of solutions will be a union ofcurves. How many? This question has been systematically studied for100 years, but only a few special cases were solved. Then, around1990, it was realized that these problems can be translated intosymplectic geometry, and then tackeled using the powerful machinery ofmathematical gauge theory. (Gauge theory, originally part of physics,has been the focus of many very fruitful interactions betweenmathematicians and physicists over the past twenty years; it includesYang-Mills and Seiberg-Witten theory, and String theory). This`Gromov invariant' approach led quickly to formulas answering some ofthe original enumerative problems, and there are clear indicationsthat there are more to be discovered. This project is aimed towardfurther developing the symplectic gauge theory in order to produceadditional general formulas, and to meld these formulas into acoherent theory.