RUI:Curve Counting Theories and Their Correspondences

枚举几何是一个经典的项目，包含的问题可以追溯到古代（有多少圆锥曲线，例如，通过五个给定点的平面？）以及当代研究的主题。尽管这些问题有着悠久的历史，但直到世纪末的一系列突破，以格罗莫夫-威滕理论的发展为高潮，许多问题才得以解决。 关键的新思想是，枚举几何的问题最好用模空间上的交集理论来解释，而弦论物理学的见解揭示了数学界以前未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的相交理论-Deligne-Mumford曲线模空间-以及为物理学家预测的一些理论之间的对应关系建立严格的数学框架。该研究计划的主要部分是致力于研究曲线模空间的Chow环。 虽然这个空间的周环的全貌目前似乎遥不可及，但有一个子环被称为“重言环”，它包含几乎所有几何上有趣的类，但承认一个明确的，有限的一组添加剂生成元。 PI与Felix Janda（密歇根大学）、Sam Grushevsky（斯托尼布鲁克大学）和Dmitry Zakharov（中密歇根大学）合作，对这些发电机之间的关系研究做出了积极贡献。 在即将到来的工作中，她计划发展重言式交叉理论的计算（生产算法计算某些理想的表达式重言式类的发电机）和理论（寻求新的重言式表达式类，如超椭圆轨迹）。 这项工作的大部分将与学生研究人员合作进行。PI研究的第二个核心组成部分是物理学提出的数学证明和等效性的扩展。 例如，在与Felix Janda和Yongbin Ruan（密歇根大学）的合作中，她证明了Gromov-Witten理论与准映射理论之间的一个跨壁公式，这种关系与著名的镜像对称物理现象密切相关。 在未来的工作中，她将把这个穿墙公式扩展到新的情况，并用它来攻击另一个物理猜想：朗道-金兹伯格/卡拉比-丘对应。 她还将与亚历山大·布里亚克合作，（利兹大学）和兰·特斯勒（Ran Tessler）（ETH苏黎世），朝着理论发展的一个版本的r-自旋理论的曲线与边界，最终目标是推广维滕的r-该奖项反映了NSF法定使命，并通过使用基金会的知识价值和更广泛的影响进行评估，被认为值得支持审查标准。

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.