Seiberg-Witten and Gromov invariants of symplectic 4-orbifolds and some applications

DMS-0304956陈伟民首席研究员计划继续他在规范理论和辛低维拓扑学方面的工作。更具体地说，他计划研究以下问题：(1)辛4-流形的辛同构群的一般性质；(2)如果6维流形上的辛圆作用是否是哈密尔顿的，如果它的不动点集是非空的和有限的；(3)代数曲面上商奇点出现的一般全局限制。在这些研究中，一个统一的主题是关于辛四维流形的Seiberg-Witten和Gromov不变量的理论，该理论将建立在Taube关于辛四维流形的相关基础工作的基础上，并实质上扩展了Taube关于辛四维流形的相关基础工作。四维orabold是一个局部看起来像我们所有人生活的时空的空间，只是局部出现了有限的对称性导致了一定程度的模棱两可。这种空间的许多例子可以在数学(例如拓扑和几何)以及理论物理的模型(例如奥比霍尔德弦理论)中很容易和自然地找到。这项拟议的研究试图理解4维空间中分布在整个空间的特定类型的场和被浓缩在2维子空间中的空间中的弦集合消除的特定类型的世界表之间的基本二元性。当有限对称不存在这样的模糊性时，这样的二元性是很好地理解的，并且是数学的一个基本部分。

DMS-0304956Weimin ChenThe Principal Investigator plans to continue his work in gaugetheory and symplectic, low-dimensional topology. More concretely,he plans to study the following problems: (1) general properties offinite groups of symplectomorphisms of a symplectic 4-manifold,(2) whether a symplectic circle action on a 6-dimensional manifoldis Hamiltonian if its fixed-point set is non-empty and finite, and(3) general, global restrictions on the occurrence of quotient singularitiesin an algebraic surface. A unifying theme in these studies is a theory,yet to be developed in this project, on the Seiberg-Witten and Gromovinvariants of a symplectic 4-orbifold, which will be built on andsubstantially extend the related fundamental work of Taubes onsymplectic 4-manifolds.A 4-dimensional orbifold is a space which locally looks like the space-time we all live in, except that there is a certain degree of ambiguity caused bya finite symmetry occured locally. Many examples of this kind of space can befound easily and naturally in Mathematics (e.g. topology and geometry) as wellas models in Theoretical Physics (e.g. orbifold string theory). The proposedresearch seeks to understand a fundamental duality in a 4-dimensional orbifoldbetween a certain type of fields that are distributed all over the space and a certaintype of worldsheet that is wiped out by a collection of strings in the space whichis condensed in a 2-dimensional subspace. When there is no such ambiguity of finite symmetries, such a duality is well-understood, and is a fundamental piece ofmathematics.