CAREER: New methods in curve counting

在过去的30年里，纯数学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，物理预言导致了数学不变量的发展，这些不变量可以计算空间中的代数曲线，相反，对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“准映射”及其组合，目标是在物理学的挑战性成果方面取得进展，直到最近数学家才达到这一目标。该项目将为研究生和博士后提供充足的培训机会。此外，PI还将每年组织一次关于感兴趣主题的周末强化学习研讨会，并组织旨在反对STEM中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，这对该技术的应用至关重要。此外，有效的不变量，这是本地化公式的主要成分，将进行研究。在另一个方向，PI将探索对数GLSM在重言式环中的应用，以建立物理学中观察到的结构预测，例如五次三重和其他单参数Calabi-Yau三重的“conifold间隙条件”，并建立所有属中五次三重的Landau-Ginzburg/Calabi-Yau对应。关于准映射，在这个项目中采用的第二个主要技术，PI将使用准映射显式计算非凸完全相交的Gromov-Witten不变量。准地图似乎需要接近一些更神秘的预测，从物理学，因此，日志GLSM将扩展到允许准地图。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.