Pseudoholomorphic Invariants of Contact Manifolds

辛结构和接触结构首先出现在经典机械系统的研究中，使人们能够描述复杂系统（如行星运动）的时间演化。在数学中，具有这些附加结构的某些几何形状被称为辛流形或接触流形。经典机械系统的解可以分别用辛流形或接触流形上的哈密顿矢量场或里布矢量场的数学对象的流线来解释。理解这些系统的演化和区别转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计划以她的工作为基础，提供接触不变量的基础和应用。PI将继续并扩大她的努力，以增加在纯数学和应用数学方面代表性不足的学生的机会和成功。她计划与莱斯大学女性数学分会合作，在莱斯大学组织每周一次的数学之夜。她还计划（与其他两位教授一起）进行两项全国性研究，以描述STEM领域学术咨询中的反种族主义形式，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，并构建一套最佳实践和有效的学术咨询行为。为了培养下一代，PI将共同组织两次国际会议，为初级数学家提供专业发展规划，并将为本科生提供研究机会。她将继续指导博士生和博士后研究人员。本课题研究基于接触流形和辛流形的伪全纯曲线的花理论不变量。这些伪全纯曲线是非线性Cauchy-Riemann方程的等价解类，它们在哈密顿向量场或Reeb向量场的闭合周期轨道之间进行插值。PI将采用直接几何方法来扩展伪全纯曲线相关模空间的横向性理论，同时主要通过使用阻塞束胶合方法来隔离和解释“错误”现象。本项目的主要目标是提供基础和完善非等变和（圆）等变接触同构的结构方面，包括发展产品结构和辛同构。通过探索阻塞束胶合的贡献，她还计划为某些闭合接触3流形中的Legendrian接触同源性提供基础。该项目的次要目标是通过开发Seifert纤维空间的嵌入式接触同源性的计算方法，为动力学、自由环路空间和辛嵌入问题提供应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.