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。其中之一是对代数曲线复杂性质的深入追求——其形式继承了高斯、阿贝尔、雅可比、黎曼、克莱因和庞加莱的作品。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义概念景观，通常与汉密尔顿、麦克斯韦、吉布斯、庞加莱、希尔伯特、爱因斯坦和魏尔的名字联系在一起。正是弦理论在探索自然的终极法则时，将代数曲线置于现代基础物理学的中心，产生了新的数学问题，并以惊人的速度和毅力指出了合理的答案。我们研究的一些问题就是由这些问题引起的，而另一些问题则有望提供弦理论学家没有预料到的答案。该奖项将用于支持至少三名博士生。PI的工作还将通过参与K-12和更高层次的教育项目，如数学奥林匹克竞赛、数学圈、图书出版和说说性写作，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和其他研究人员共同开发，并基于这些流形中全纯曲线模空间上向量束的性质研究哈密顿系统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，以及它们的上同调的前身，应该是更一般的协值不变量的专门化。一种方法，在这里被称为“形式”，忽略了模空间的微妙的堆叠或轨道特性，应该与充分捕捉这些微妙之处的“真正”理论进行对比。在这个项目中，PI和他的合作者将超越上同调和k理论的Gromov-Witten不变量，并探索定义和计算真正的不变量的可能性，而不是形式的，在复杂协数中具有值的Gromov-Witten不变量。一个方向是包括量子“chi-y”理论（基于Hirzebruch属），与预期的联系和应用于表示理论和颤振变种。另一个方向是探索基于稳定映射模空间的拓扑欧拉特征对理论（y=-1）的专门化。另一个是考察基于椭圆上同调的Gromov-Witten理论的前景。背景上的一般问题是：哪些非凡的上同调理论可以包含所有紧复轨道（与流形相反）？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。

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.