Topology of smooth and symplectic 4-manifolds

光滑和辛4流形的拓扑

• 批准号: 1510395
• 负责人: R. Inanc Baykur
• 金额: $ 17.86万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510395&HistoricalAwards=false
• 关键词: Topology; smooth; symplectic; manifolds

As our world, with time included, is four dimensional, exploring similarities and differences of 4-dimensional spaces through geometric and topological methods leads to better understanding of the universe we live in. Smooth and symplectic structures on 4-dimensional spaces are broadly featured in theoretical physics; e.g. in classical mechanics, string and quantum field theories. The research projects invoke new ideas and techniques, creating leverage to address several important problems regarding the topology of smooth and symplectic 4-manifolds, and that of contact 3-manifolds and their fillings. A key aspect of this program is the reduction of many intricate problems to fairly simple algebraic relations between curves on surfaces, which sets an excellent ground to present problems accessible to graduate and advanced undergraduate students. The PI will engage and mentor students in related research topics.This is a project in low dimensional geometry and topology, focusing on a variety of profound questions involving symplectic 4-manifolds and contact 3-manifolds. Some of the projects on 4-manifolds are pertinent to classification of symplectic Calabi-Yau surfaces, diversity of Lefschetz pencils and fibrations, novel constructions of small exotic and symplectic 4-manifolds, and complexity in various stable equivalences. As for 3-manifolds, the PI will probe the complexity in Giroux correspondence in terms of open book support genus, the characterization of newly discovered contact 3-manifolds with arbitrarily large Stein fillings, diversity of complex curves bounding links in the 3-sphere, and generalizations of quasi-positive links. Gauge theory, new mapping class group techniques and symplectic surgeries will play a vital role in this program.

随着时间的流逝，我们的世界是四个维度，通过几何和拓扑方法探索4维空间的相似性和差异，可以更好地理解我们所生活的宇宙。在理论物理学中，四维空间上的平滑和符号结构在4维空间中广泛特征；例如在经典力学，弦和量子场理论中。研究项目引用了新的想法和技术，从而杠杆作用，以解决有关平滑和符号4个manifolds拓扑的几个重要问题，以及联系3个manifolds及其填充物的拓扑。该程序的一个关键方面是将许多复杂问题减少到表面上曲线之间相当简单的代数关系，这为研究生和高级本科生而言可以解决问题。 PI将吸引并指导学生参与相关研究主题。这是一个低维几何和拓扑的项目，重点是涉及涉及Symblectic 4-Manifolds和联系3个Manifolds的各种深刻的问题。 4个manifolds上的某些项目与符号成绩的calabi-yau表面，lefschetz铅笔的多样性和纤维的分类有关，这是各种稳定等价方面的小型外来和同型4个manifords的新颖结构，以及复杂性的复杂性。至于3个体，PI将根据开放式书籍支持属探测Giroux对应关系的复杂性，新发现的触点3型触点具有任意较大的Stein填充物，复杂曲线界限的多样性，在3-Sphere中的多样性以及Quasi-Post-persitive链接的概括。量规理论，新的映射类小组技术和符号手术将在该计划中起着至关重要的作用。