Pseudoholomorphic Invariants of Contact Manifolds

辛结构和接触结构最早出现在对经典力学系统的研究中，使人们能够描述复杂系统的时间演化，如行星运动。在数学中，具有这些附加结构的某些几何形状称为辛流形或接触流形。经典力学系统的解可以用数学对象的流线来解释，分别称为辛流形或接触流形上的哈密顿矢量场或里布矢量场。了解这些系统的演化和区分变换导致了辛结构和接触结构的全局不变量的发展，这揭示了动力学、几何和拓扑的相互联系。PI计划在她工作的基础上提供接触不变量的基础和应用。国际数学联合会将继续并扩大她的努力，以增加未被充分代表的学生在理论数学和应用数学方面的机会和成功。与莱斯女性数学分会合作，她计划在莱斯大学组织每周一次的数学之夜。她还计划(与另外两位教授)进行两项全国性研究，以描绘STEM领域学术咨询中反种族主义的形式，并研究学术顾问的做法对BIPOC(黑人、土著和有色人种)学生心理体验和结果的影响，并构建一套学术建议的最佳实践和有效行为。为了培养下一代，国际数学家协会将为初级数学家联合举办两个具有专业发展规划的国际会议，并将为本科生提供研究机会。她将继续为博士生提供建议，并指导博士后研究人员。该项目涉及基于拟全纯曲线的接触流形和辛流形的Floer理论不变量。这些伪全纯曲线是非线性Cauchy-Riemann方程的解的等价类，该方程在哈密顿向量场或Reeb向量场的闭合周期轨道之间插补。PI将使用直接几何方法来扩展伪全纯曲线的相关模空间的横截性理论，同时主要通过使用阻塞束粘合方法来隔离和解释“错误”现象。这个项目的主要目标是提供基础和完善非等变和(圆)等变接触同调的结构方面，包括开发具有辛同调的乘积结构和同构。通过探索障碍束粘合的贡献，她还计划为某些封闭接触3-流形中的Legendrian接触同调提供基础。这个项目的次要目标是提供动力学、自由回路空间和辛嵌入问题的应用，部分是通过开发Seifert纤维空间的嵌入接触同调的计算方法。该奖项反映了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。

Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of complex systems such as planetary motion. In mathematics, certain geometric shapes with these additional structures are known as symplectic or contact manifolds. Solutions of classical mechanical systems can be interpreted in terms of flow lines of mathematical objects known as Hamiltonian or Reeb vector fields on a symplectic or contact manifold, respectively. Understanding the evolution and distinguishing transformations of these systems led to the development of global invariants of symplectic and contact structures, which shed light on the interconnectedness of dynamics, geometry, and topology. The PI plans to build on her work in providing foundations and applications of contact invariants. The PI will continue and expand her efforts to increase the access and success of underrepresented students in pure and applied mathematics. With the Rice Association for Women in Mathematics Chapter, she plans to organize a weekly Math Night at Rice University. She also plans to conduct (jointly with two other professors) two national studies to delineate forms of antiracism in academic advising in STEM fields and to examine the effect of academic advisors' practices on BIPOC (Black,Indigenous, and people of color) student psychological experiences and outcomes, and construct a set of best practices and effective behaviors in academic advising. To train future generations, the PI will co-organize two international conferences with professional development programming for junior mathematicians and will offer research opportunities for undergraduates. She will continue to advise PhD students and mentor postdoctoral researchers. The project concerns pseudoholomorphic curve based Floer theoretic invariants of contact and symplectic manifolds. These pseudoholomorphic curves are equivalence classes of solutions to a nonlinear Cauchy-Riemann equation which interpolate between closed periodic orbits of either a Hamiltonian or Reeb vector field. The PI will employ direct geometric methods to extend the transversality theory for the associated moduli spaces of pseudoholomorphic curves while isolating and accounting for “errant” phenomena, primarily through the use of obstruction bundle gluing methods. The primary goals of this project are to provide foundations and refine structural aspects of nonequivariant, and (circle) equivariant contact homology, including the development of product structures and isomorphisms with symplectic homology. By exploring contributions from obstruction bundle gluing, she also plans to provide foundations for Legendrian contact homology in certain closed contact 3-manifolds. The secondary goals of this project are to provide applications to dynamics, free loop spaces, and symplectic embedding problems, in part through developing computational methods for embedded contact homology of Seifert fiber 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.