CAREER: New methods in curve counting

在过去的三十年里，纯数学中的几个分支与物理学中的弦理论之间发生了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，相反，对这些不变量的数学研究促进了弦理论的发展。该项目进一步开发了两种曲线计数技术，即“对数规范线性西格玛模型”(LOG GLSM)和“拟映射”及其组合，目的是在挑战物理学的猜想方面取得进展，直到最近，这些猜想似乎还遥不可及。该项目将为研究生和博士后提供充足的培训机会。此外，国际和平研究所将每年举办一次关于感兴趣的专题的周末强化学习讲习班，并组织旨在对抗STEM中的陈规定型观念的活动。更具体地说，该项目将证明LOG GLSM的本地化公式，这对这项技术的应用至关重要。此外，还将研究作为本地化公式的主要成分的有效不变量。在不同的方向上，PI将探索LOG GLSM在重言环中的应用，建立物理学中观察到的五次三重和其他单参数Calabi-Yau三重的结构预测，并建立所有属的五次三重的Landau-Ginzburg/Calabi-Yau对应。关于拟映射，这是本项目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点的Gromov-Witten不变量。准地图似乎是接近物理学中一些更神秘的预测所必需的，因此LOG GLSM将扩展到允许准地图。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.