Geometry and topology of smooth four-manifolds

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。第一部分涉及产生新的光滑和辛闭四维流形，以解决各种问题，从构造具有小欧拉特征的标准四维流形的非同构副本到具有非平凡基本群的四维流形的辛地理问题，或从建立新的奇异家庭的四流形，是区别于他们的稳定同伦塞伯格-维滕不变量获得光滑纽结，但拓扑非纽结嵌入的表面。该项目的第二部分涉及光滑四流形上的Lefschetz纤维化和辛结构的推广。本文将奇异性理论与奇异体技术相结合，得到了光滑四维流形上广义纤维化的新结果，并利用断裂Lefschetz纤维化和它们之间的运动（类似于奇异体和Kirby运动）建立了光滑四维流形的一个有用的描述.确定哪些破Lefschetz纤维化可以或不可以支持具有非平凡Seiberg-Witten不变量的光滑四维流形，以及利用与之相关的破Lefschetz纤维化研究某些四维流形的同构类型是本研究的另外两个问题。该项目的目标是更好地理解“四维流形”的有趣性质，四维流形是在时空上局部建模的几何对象。在理论物理学中有大量的文献与四维流形的“形状”有关，并且有大量的数学研究致力于这个主题。当考虑某些附加结构时，四维流形表现出许多奇怪的差异。首先，出现在经典力学和弦理论中各种方程中的“辛结构”构成了拟议研究的一个关键主题。在量子场论中，由微分方程产生的“塞伯格-威滕不变量”也起着关键作用。运用几何和拓扑的方法，沿着新的结构，PI研究了四维流形的相似性和差异性。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposed research has two main parts. The first part concerns producing new smooth and symplectic closed four-manifolds so as to address a variety of problems that range from constructing non-diffeomorphic copies of standard four-manifolds with small Euler characteristics to the symplectic geography problem for four-manifolds with nontrivial fundamental groups, or from building new exotic families of four-manifolds that are distinguished by their stable cohomotopy Seiberg-Witten invariants to obtaining smoothly knotted but topologically unknotted embeddings of surfaces. The second part of the project deals with generalizations of Lefschetz fibrations and symplectic structures on smooth four-manifolds. In this research, singularity theory and handlebody techniques are combined to obtain new results on generalized fibrations on smooth four-manifolds, and to establish a useful description of smooth four-manifolds in terms of broken Lefschetz fibrations and moves between them, analogous to handlebodies and Kirby moves. Determining which broken Lefschetz fibrations can or cannot support a smooth four-manifold with nontrivial Seiberg-Witten invariant, and investigating the diffeomorphism types of certain four-manifolds using broken Lefschetz fibrations associated to them are two other problems contained in this research.Space and time combined, we live in a four dimensional world. The goal of this project is to better understand the intriguing nature of "four-manifolds", which are geometric objects locally modeled on space-time. There is an immense literature in theoretical physics related to the 'shape' of four-manifolds, and a great deal of mathematical research dedicated to this very subject. When considered with certain additional structures, four-manifolds exhibit numerous curious differences. For one, "symplectic structures", which appear in various equations in classical mechanics and string theory constitute a key theme of the proposed research. The "Seiberg-Witten invariants" that arise from differential equations in quantum field theory also play a key role. Using geometric and topological methods, along with new structures, the PI studies similarities and differences of four-manifolds.