Gromov-Witten Invariants and Extraordinary Cohomology

从一般的角度来看，我们在研究中处理的问题位于过去两个世纪数学中两条主要途径的交叉点。其中之一是深入追求的复杂性质的代数曲线-在形式继承的作品高斯，阿贝尔，雅可比，黎曼，克莱因和庞加莱。另一个是由经典力学和量子力学的进步所决定的数学物理的广泛概念景观，并经常与汉密尔顿，麦克斯韦，吉布斯，庞加莱，希尔伯特，爱因斯坦和外尔的名字联系在一起。弦理论在探索自然界的终极定律时，把代数曲线置于现代基础物理学的中心，以惊人的速度和毅力提出新的数学问题，并指出似乎合理的答案。我们研究的一些问题就是由这些问题激发的，另一些问题则希望提供弦理论家没有真正预料到的答案。该奖项将用于支持至少三名博士生。PI的工作还将通过他参与K-12和更高级别的教育项目，如数学奥林匹克，数学圈，图书出版和论文写作，影响STEM教育的氛围。 该项目的具体目标建立在所谓的K理论Gromov-Witten不变量理论的成功基础上，该理论由PI和其他研究人员开发，并基于这些流形中全纯曲线的模空间上的向量丛的性质研究Hamilton系统的相流形的拓扑不变量。从形式同伦理论的观点来看，K-理论不变量，以及它们的上同调前辈，应该是更一般的配边值不变量的特殊化。一种方法，这里被称为“正式”，忽视了微妙的堆叠或orbifold性质的模空间，并应与“真正的”理论完全捕捉这些微妙之处。 在这个项目中，PI和他的合作者将超越上同调和K理论Gromov-Witten不变量，并探索定义和计算真正的可能性，而不是正式的，Gromov-Witten不变量与复杂的协边值。一个方向是包括量子“chi-y”理论（基于Hirzebruch属），预期与表示论和量子力学的应用有联系。 另一个方向是探索基于稳定映射的模空间的拓扑欧拉特征的理论（其中y=-1）的特殊化。另一个是考察基于椭圆上同调的Gromov-Witten理论的发展前景。背景上的一般问题是：哪些非凡的上同调理论可以家所有的紧凑复杂orbifolds（相对于流形）？该奖项反映了NSF的法定使命，并被认为是值得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的支持。

From a general perspective, problems we deal with in our research lie at the intersection of two major pathways in mathematics of the past two centuries. One of them is the in-depth pursuit of the intricate properties of algebraic curves - in the form inherited from works of Gauss, Abel, Jacobi, Riemann, Klein and Poincare. The other is the broad conceptual landscaping of mathematical physics dictated by the progress of classical and quantum mechanics, and often associated with the names of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einstein and Weyl. It is string theory that in the search for the ultimate laws of nature places algebraic curves at the center of modern fundamental physics, generating new mathematical questions and pointing out plausible answers with an amazing pace and persistence. Some of the problems we work on are motivated by such questions, some others hopefully provide answers that string theorist did not really anticipate. The award will be used to support at least three PhD students. The PI's work will also influence the climate in STEM education through his involvement with K-12 and higher level educational projects, such as Math Olympiads, math circles, book publishing, and expository writing. The specific goals of the project build upon the success of the theory of the so-called K-theoretic Gromov-Witten invariants, developed by the PI among other researchers, and studying topological invariants of phase manifolds of Hamiltonian systems based on properties of vector bundles over moduli spaces of holomorphic curves in these manifolds. From a formal homotopy theory viewpoint, K-theoretic invariants, as well as their cohomological predecessors, should be specializations of much more general cobordism-valued invariants. An approach, referred here as "formal", disregards the subtle stacky or orbifold properties of the moduli spaces, and should be contrasted with the "genuine" theory fully capturing these subtleties. In this project, the PI and his collaborators will move beyond cohomological and K-theoretic Gromov-Witten invariants, and explore the possibility of defining and computing genuine, as opposed to formal, Gromov-Witten invariants with values in complex cobordisms. One direction is to include quantum "chi-y"-theory (based on the Hirzebruch genus), with expected connections with, and applications to representation theory and quiver varieties. Another direction is to explore the specialization to the theory (where y=-1) based on the topological Euler characteristics of moduli spaces of stable maps. Yet another one is to examine the prospects of Gromov-Witten theory based on elliptic cohomology. The general question on the background is: Which extraordinary cohomology theories can home all compact complex orbifolds (as opposed to manifolds)?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.