CAREER: Symplectic and Holomorphic Convexity in 4-dimensions

拓扑学研究不同空间的形状，例如DNA链、大型数据集，甚至宇宙。特别是在拓扑学中要研究的自然空间是所谓的光滑流形。平滑流形的一个例子是我们生活的宇宙，我们认为它是三维的，但如果我们包括时间，实际上是四维的。一个基本的问题是：我们的宇宙是什么流形？几何拓扑学家的目标是列出可能性，最重要的是开发工具来理解流形的结构和区分流形的整体拓扑。低维拓扑主要是指对三维和四维流形的研究，其中关于这种空间的理论，特别是在四维空间中，要复杂得多，并且表现出许多在其他任何维度上都没有见过的独特现象。理解流形拓扑的一个特别成功和丰富的方法是在流形上引入某些复杂的解析结构，如Stein结构和/或几何结构，如辛几何和接触几何。PI将研究位于低维的辛拓扑/接触拓扑学和光滑流形拓扑学以及复杂几何的交叉处的问题和猜想。除了研究部分，该项目还包括将PI的研究计划与研究生和本科生研究的教育和培训倡议相结合的活动。为此，PI将通过组织研讨会、研讨会、REU和会议，在阿拉巴马大学保持一个活跃的拓扑组。更广泛地说，PI将利用这一奖项的资金，为阿拉巴马州代表性不足的大学的有才华的本科生组织为期一周的研究和专业发展夏季研讨会。他还将为东南会议(SEC)学校的几何拓扑学研究生和教职员工启动一个合作的、多维度的长期教育和研究计划，最初包括阿拉巴马大学、路易斯安那州立大学、密西西比大学和阿肯色大学。这个项目涉及研究复杂几何、辛几何和接触几何之间的相互作用，以及低维拓扑之间的相互作用，重点是各种凸性概念。第一个长期的研究目标是从辛拓扑(例如接触型超曲面)和复几何(例如全纯/有理/多项式凸Stein域的边界)的角度考虑嵌入在4-空间中的闭3-流形的拓扑约束。具体地说，PI将致力于完全解决Gompf的猜想，该猜想预测在复2-空间中没有Brieskorn球界全纯凸域，确定4-空间和复射影空间中接触型超曲面的拓扑，并探索它们对光滑4-流形拓扑的奇异性质的影响。第二个长期研究目标是关于封闭定向三维流形上紧密可填充接触结构的存在和分类的两个突出问题。PI的各种节理工作为从Heegaard-Floer同调的角度研究紧密可填充接触结构的存在问题提供了一个框架和新的构造。PI和他的合作者将调查来自Heegaard Floer同源的不变量在多大程度上完全检测紧密性。PI还将致力于完成小Seifert纤维空间上紧密结构的分类问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is the study of the shape of different spaces, examples of which include a strand of DNA, a large data set, or even the universe. Particularly natural spaces to study in topology are the so called smooth manifolds. An example of a smooth manifold is the universe we live in, which we think of as 3-dimensional, but if we include time is actually 4-dimensional. A fundamental question is: what manifold is our universe? A goal of a geometric topologist is to list possibilities and most importantly to develop tools to understand the structure and distinguish the overall topology of manifolds. Low dimensional topology mainly refers to the study of 3-and 4-dimensional manifolds, where the theory of such spaces, in particular in 4-dimension, is significantly more complicated and exhibits many unique phenomena that are not seen in any other dimension. A particularly successful and rich way to understand the topology of manifolds is to introduce certain complex analytic structures on them, such as Stein structures and/or geometric structures, such as symplectic and contact geometries. The PI will study questions and conjectures that sit at the intersection of symplectic/contact topology and smooth manifold topology in low dimensions, and complex geometry. Alongside the research component, this project also includes activities that integrate the PI’s research program with education and training initiatives for student research, at both graduate and undergraduate levels. To that end, the PI will maintain an active topology group at the University of Alabama by organizing seminars, workshops, REUs and conferences. More broadly, the PI will use funds from this award to organize a week-long research and professional development summer workshop for talented undergraduate students from underrepresented colleges in the state of Alabama. He will also initiate a collaborative, multi-dimensional, long-term educational and research program for geometry-topology graduate students and faculty in the Southeastern Conference (SEC) schools, initially including the University of Alabama, Louisiana State University, the University of Mississippi, and the University of Arkansas. This project involves studying interactions between complex geometry, symplectic geometry and contact geometry, and low dimensional topology with a focus on various notions of convexity. The first long-term research objective will be to consider constraints on the topology of closed 3-manifolds embedded in 4-space from the perspectives of symplectic topology (e.g. contact type hypersurfaces) and complex geometry (e.g. the boundaries of holomorphically/rationally/polynomially convex Stein domains). Specifically, the PI will work towards a complete resolution of Gompf’s conjecture that predicts no Brieskorn sphere bounds a holomorphically convex domain in complex 2-space, determining the topology of contact type hypersurfaces in 4-space and in complex projective space, and exploring their implications for the exotic nature of smooth 4-manifold topology. The second long-term research objective concerns two outstanding problems dealing with the existence and classification of tight and fillable contact structures on closed, oriented 3-manifolds. The PI’s various joint works provide a framework and new constructions to study the existence question of tight and fillable contact structures from the point of view of Heegaard Floer homology. The PI and his collaborators will investigate to what extent invariants coming from Heegaard Floer homology detect tightness completely. The PI will also work to complete the classification problem for tight structures on small Seifert fibered spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.