Topology of smooth and symplectic 4-manifolds

由于包括时间在内的我们的世界是四维的，通过几何和拓扑方法探索四维空间的异同可以更好地理解我们生活的宇宙。4维空间上的光滑和辛结构在理论物理中具有广泛的特征，例如在经典力学、弦和量子场论中。这些研究项目引入了新的思想和技术，创造了杠杆来解决关于光滑和辛四维流形的拓扑以及接触三维流形及其填充的几个重要问题。本课程的一个关键方面是将许多复杂的问题归结为曲面上曲线之间的相当简单的代数关系，这为研究生和高级本科生提出问题奠定了良好的基础。这是一个关于低维几何和拓扑学的项目，专注于涉及辛4流形和接触3流形的各种深奥的问题。4-流形上的一些项目涉及到辛Calabi-Yau曲面的分类，Lefschetz铅笔和纤维的多样性，小型奇异和辛4-流形的新构造，以及各种稳定等价的复杂性。对于3-流形，PI将从开书支撑亏格、新发现的具有任意大Stein填充的接触3-流形的刻画、3-球面中复杂曲线有界环的多样性以及拟正环的推广等方面探讨Giroux对应的复杂性。规范理论、新的映射类群技术和辛外科手术将在这一计划中发挥重要作用。

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.