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.

由于我们的世界，包括时间，是四维的，通过几何和拓扑方法探索四维空间的相似性和差异性，可以更好地了解我们生活的宇宙。四维空间上的光滑和辛结构在理论物理学中有广泛的特点，例如在经典力学、弦理论和量子场论中。这些研究项目引用了新的思想和技术，创造了解决光滑和辛4-流形拓扑结构以及接触3-流形及其填充的几个重要问题的杠杆作用。该计划的一个关键方面是减少了许多复杂的问题，以相当简单的曲面上的曲线之间的代数关系，这为研究生和高级本科生提供了一个很好的基础。PI将参与并指导学生相关的研究课题。这是一个低维几何和拓扑学的项目，专注于涉及辛4-流形和接触3-流形的各种深刻问题。4-流形上的一些项目涉及辛卡-丘曲面的分类、莱夫谢茨铅笔和纤维化的多样性、小奇异和辛4-流形的新颖构造以及各种稳定等价的复杂性。至于3-流形，PI将探讨Giroux对应的复杂性，包括开卷支撑亏格，新发现的具有任意大Stein填充的接触3-流形的特征，3-球面中复杂曲线边界链接的多样性，以及准正链接的推广。规范理论，新的映射类组技术和辛手术将在这个计划中发挥至关重要的作用。