Moduli Problems in Contact Geometry

辛结构和接触结构首先出现在经典力学系统的研究中，允许人们描述简单和复杂系统的时间演化，如弹簧，行星运动和波传播。 经典力学中的运动方程是由一个守恒量，即能量的概念决定的。 一个相关的量是作用量，它通过运动方程的解而被最小化。 对于一个封闭系统，例如开普勒问题，其解描述了行星绕太阳运行的路径，能量是系统中动能和势能的总和，作用量是动能减去势能的（最小化）平均值。 当我们通过解开隐藏在能量和作用概念中的信息来研究这些系统时，辛结构和接触结构就出现了。 理解这些系统的演化和区别变换导致了辛流形和接触流形的全局不变量的发展。 PI计划继续她的工作，提供基础和应用程序的接触不变量源于nonequivariant和（圆）equivariant建设的接触同源性。接触同调建立了封闭的Reeb向量场的轨道和计数的解决方案的非线性柯西-黎曼方程之间的封闭Reeb轨道插值。Reeb向量场是类哈密顿向量场，其流线是汉密尔顿运动方程的解，因为它们保存能量。闭合Reeb轨道特别令人感兴趣，因为它们可以用来描述行星绕星星的运动，卫星的闭合轨道，以及其他局部距离最小化“回路”。“接触同调可以用来更好地区分给定光滑流形上的接触结构，并提取与可能的Reeb向量场有关的动力学信息，这些向量场可以与固定接触结构相关联。PI的工作表明，这些不变量捕获的现象比以前预期的要多。她还计划研究这些接触和相关的辛不变量，勒让德结在封闭的接触流形，塞弗特纤维空间的动力学和辛嵌入的其他结构特性。PI计划继续她的外展计划，增加了代表性不足的学生和教师在数学方面的机会和成功，并在当地高中生和本科生中培养了对几何和拓扑学的更大兴趣。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex systems such as springs, planetary motion, and wave propagation. The equations of motion in classical mechanics are determined by the notion of a conserved quantity, energy. A related quantity is action, which is minimized by solutions to the equations of motion. For a closed system, such as the Kepler problem whose solutions describe paths of planets orbiting the sun, the energy is the sum of the kinetic and potential energy in the system, and the action is given by the (minimized) mean value of kinetic minus potential energy. Symplectic and contact structures emerge as we investigate these systems by unpacking the information hidden in the notions of energy and action. Understanding the evolution and distinguishing transformations of these systems led to the development of global invariants of symplectic and contact manifolds. The PI plans to continue her work in providing foundations and applications for contact invariants stemming from nonequivariant and (circle) equivariant constructions of contact homology. Contact homology is built out of closed orbits of Reeb vector fields and counts of solutions to a nonlinear Cauchy-Riemann equation which interpolates between closed Reeb orbits. Reeb vector fields are Hamiltonian-like vector fields, whose flow lines are solutions to Hamilton's equations of motion, as they conserve energy. Closed Reeb orbits are of particular interest because they can be used to describe the motion of a planet orbiting a star, the closed trajectories of a satellite, and other local distance minimizing "loops." Contact homology can be used to better distinguish contact structures on a given smooth manifold and to extract dynamical information pertaining to the possible Reeb vector fields which can be associated to a fixed contact structure. The PI's work has shown that these invariants capture more phenomena than previously expected. She also plans to study additional structural properties of these contact and related symplectic invariants, Legendrian knots in closed contact manifolds, dynamics of Seifert fiber spaces, and symplectic embeddings. The PI plans to continue her outreach programs which have increased the access and success of underrepresented students and faculty in mathematics as well as foster greater interest in geometry and topology amongst local high school students and undergraduates.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.