CAREER: New methods in curve counting

在过去的三十年里，纯数学中的几个分支和物理学中的弦理论之间有着深刻而令人惊讶的相互作用。特别是，物理预测导致了计算空间中代数曲线的数学不变量的发展，反过来，这些不变量的数学研究导致了弦理论的进步。该项目进一步发展了两种曲线计数技术，即“对数测量线性西格玛模型”（log GLSM）和“准映射”，以及它们的组合，目标是在挑战物理学中的猜想方面取得进展，这些猜想直到最近才出现在数学家的范围之外。本项目将为研究生和博士后提供充足的培养机会。此外，PI将组织每年一次关于感兴趣主题的周末密集学习研讨会，并组织旨在消除STEM刻板印象的活动。更具体地说，该项目将得到测井GLSM定位公式的证明，这对该技术的应用至关重要。此外，还将研究局部化公式的主要组成部分——有效不变量。在不同的方向上，PI将探索对数GLSM在同音环中的应用，建立物理上观察到的结构预测，如五次三次和其他单参数Calabi-Yau三次的“confold gap条件”，并建立所有属的五次三次的Landau-Ginzburg/Calabi-Yau对应关系。关于拟映射，在这个项目中采用的第二种主要技术，PI将使用拟映射来显式计算非凸完全交的Gromov-Witten不变量。准映射对于接近一些来自物理学的更神秘的预测似乎是必要的，因此log GLSM将被扩展以允许准映射。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。

The past thirty years have seen a deep and surprising interplay between several branches in pure mathematics, and string theory in physics. In particular, physical predictions have led to the development of mathematical invariants which count algebraic curves in spaces, and conversely, the mathematical study of these invariants has led to advances in string theory. This project further develops two curve counting techniques, the "logarithmic gauged linear sigma model" (log GLSM) and "quasimaps", and their combination, with the goal of making progress on challenging conjectures from physics, which have appeared out of reach of mathematicians until recently. This project will offer ample training opportunities for graduate students and postdocs. In addition, the PI will organize a yearly intensive weekend learning workshop on a topic of interest, as well as organize events aiming to counter stereotypes in STEM.More specifically, the project will result in a proof of the localization formula for log GLSM, which is of utmost importance for the application of this technique. In addition, effective invariants, which are a major ingredient of the localization formula, will be studied. In a different direction, the PI will explore applications of log GLSM to the tautological ring, to establish structural predictions observed in physics, such as the "conifold gap condition", for the quintic threefold and other one-parameter Calabi-Yau threefolds, and to establish the Landau-Ginzburg/Calabi-Yau correspondence for quintic threefolds in all genera. With regard to quasi-maps, the second main technique employed in this project, the PI will use quasi-maps for explicit computations of Gromov-Witten invariants of non-convex complete intersections. Quasi-maps appear necessary for approaching some of the more mysterious predictions from physics, and hence log GLSM will be extended to allow for quasi-maps.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.