RUI:Curve Counting Theories and Their Correspondences

RUI：曲线计数理论及其对应关系

• 批准号: 1810969
• 负责人: Emily Clader
• 金额: $ 18.9万
• 依托单位: San Francisco State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810969&HistoricalAwards=false
• 关键词: RUI; Curve; Counting; Theories; Their

Enumerative geometry is a classical project, encompassing questions that date back to antiquity (how many conics, for example, pass through five given points in the plane?) as well as subjects of contemporary research. Despite their long history, many of these questions remained inaccessible until a series of breakthroughs in the late twentieth century that culminated in the development of Gromov-Witten theory. The key new ideas were that problems of enumerative geometry are best interpreted in terms of intersection theory on a moduli space, and that insights from the physics of string theory reveal elegant patterns in families of enumerations that were previously unobserved by the mathematics community. The PI's research program involves developing the intersection theory of a particularly fundamental moduli space - the Deligne-Mumford moduli space of curves - as well as producing rigorous mathematical frameworks for some of the correspondences between theories that physicists have predicted.A major part of this research program is devoted to studying the Chow ring of the moduli space of curves. Although a full picture of the Chow ring of this space currently seems out of reach, there is a subring known as the "tautological ring" that contains nearly every geometrically-interesting class yet that admits an explicit, finite set of additive generators. The PI has been an active contributor to the study of the relations among these generators, in joint work with Felix Janda (University of Michigan), Sam Grushevsky (Stony Brook University), and Dmitry Zakharov (Central Michigan University). In forthcoming work, she plans to develop tautological intersection theory both computationally (producing algorithms for calculating certain desirable expressions for tautological classes in terms of the generators) and theoretically (seeking new tautological expressions for classes such as the hyperelliptic locus). Much of this work will be carried out in collaboration with student researchers. A second central component of the PI's research is the mathematical proof and extension of equivalences proposed by physics. For example, in joint work with Felix Janda and Yongbin Ruan (University of Michigan), she has proven a wall-crossing formula relating Gromov-Witten theory to the theory of quasimaps, a relationship that is closely related to the famous physical phenomenon known as mirror symmetry. In future work, she will extend this wall-crossing formula to new cases and use it to attack another physical conjecture: the Landau-Ginzburg/Calabi-Yau correspondence. She will also work, in collaboration with Alexandr Buryak (University of Leeds) and Ran Tessler (ETH Zurich), toward the theoretical development of a version of r-spin theory for curves with boundary, with an ultimate goal of generalizing Witten's r-spin conjecture to that setting.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.

列举几何是一个经典的项目，包含了可以追溯到古代的问题(例如，有多少二次曲线通过平面上的五个给定点？)以及当代研究的主题。尽管这些问题有很长的历史，但许多问题直到20世纪末的一系列突破才得以解决，这些突破最终导致了格罗莫夫-维滕理论的发展。关键的新想法是，枚举几何问题最好地用模空间上的交集理论来解释，来自弦理论物理学的见解揭示了数学界以前没有观察到的枚举族中的优雅模式。PI的研究计划包括发展一个特别基本的模空间--Deligne-Mumford模空间的交集理论，以及为物理学家预测的一些理论之间的对应建立严格的数学框架。这个研究计划的主要部分致力于研究曲线模空间的Chow环。虽然这个空间的Chow环的全貌目前看起来还不太可能，但有一个被称为“重言环”的子环，它几乎包含了每一个几何有趣的类，它允许一个显式的，有限的加性生成元集合。国际和平研究所与Felix Janda(密歇根大学)、Sam Grushevsky(石溪大学)和Dmitry Zakharov(中密歇根大学)共同工作，为研究这些发电机之间的关系作出了积极贡献。在接下来的工作中，她计划发展重言式交集理论，既在计算上(产生算法，根据生成器计算重言式类的某些理想表达式)，也在理论上(为超椭圆轨迹等类寻找新的重言式表达式)。这项工作的大部分将与学生研究人员合作进行。PI研究的第二个核心部分是由物理学提出的等价性的数学证明和扩展。例如，在与密歇根大学的菲利克斯·扬达和阮永斌的合作中，她证明了一个将Gromov-Witten理论与准映射理论联系起来的跨墙公式，这种关系与著名的被称为镜像对称的物理现象密切相关。在未来的工作中，她将把这个跨越墙的公式推广到新的案例，并用它来破解另一个物理猜想：兰道-金兹堡/卡拉比-尤对应关系。她还将与利兹大学的Alexandr Buryak和苏黎世ETH的Ran Tessler合作，致力于为有边界的曲线开发一个版本的r-自旋理论，最终目标是将Witten的r-自旋猜想推广到该背景下。该奖项反映了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。

项目成果

Open 𝑟-Spin Theory I: Foundations

• DOI: 10.1093/imrn/rnaa345
• 发表时间: 2021-02
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: A. Buryak;E. Clader;Ran J. Tessler

Wall-crossing in genus-zero hybrid theory

• DOI: 10.1515/advgeom-2021-0010
• 发表时间: 2018-06
• 期刊: Advances in Geometry
• 影响因子: 0.5
• 作者: E. Clader;Dustin Ross

Higher-genus wall-crossing in the gauged linear sigma model

• DOI: 10.1215/00127094-2020-0053
• 发表时间: 2017-06
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: E. Clader;F. Janda;Y. Ruan

Boundary complexes of moduli spaces of curves in higher genus

高阶曲线模空间的边界复形

• DOI: 10.1090/proc/15423
• 发表时间: 2022
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Clader, Emily;Luber, Dante;Quillin, Kyla
• 通讯作者: Quillin, Kyla