CAREER: New methods in curve counting

职业：曲线计数的新方法

基本信息

批准号：
2422291
负责人：
Felix Janda
金额：
$ 41.71万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2024
资助国家：
美国
起止时间：
2024-03-01 至 2028-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2422291&HistoricalAwards=false
关键词：
CAREER New methods curve counting

项目摘要

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.

过去三十年来，纯数学中的几个分支与物理学的弦理论之间看到了深刻而令人惊讶的相互作用。特别是，物理预测导致了数学不变性的发展，这些数学不变性数在空间中计算代数曲线，相反，这些不变性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“对数测量的线性sigma模型”（LOOD GLSM）和“ Quasimaps”及其组合，目的是在物理学的挑战性猜想上取得进展，这些物理学的挑战性猜想直到最近才出现在数学家的范围内。该项目将为研究生和博士后提供充足的培训机会。此外，PI将组织一个有关感兴趣的主题的年度强化周末学习研讨会，并组织旨在对抗STEM中的刻板印象的事件。更具体地说，该项目将证明Log GLSM的本地化公式，这对于应用这项技术至关重要。此外，将研究有效的不变性，这是本地化公式的主要成分。 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.关于准映射，该项目采用的第二个主要技术，PI将使用准映射来明确计算非convex完整交叉点的Gromov-witten不变性。对于接近物理学的一些更神秘的预测似乎是必要的，因此将扩展Log GLSM以允许进行准映射。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的审查标准来通过评估来进行评估的。