RUI:Curve Counting Theories and Their Correspondences

枚举几何是一项经典工程，涵盖了可以追溯到古代的问题（例如，有多少条圆锥曲线经过平面上的五个给定点？）以及当代研究的主题。尽管这些问题有着悠久的历史，但直到20世纪后期格鲁莫夫-威滕理论的发展取得了一系列突破，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，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的优雅模式。PI的研究项目包括发展一个特别基本的模空间的相交理论——曲线的Deligne-Mumford模空间——以及为物理学家预测的理论之间的一些对应关系建立严格的数学框架。本课题主要研究曲线模空间的周氏环。虽然这个空间的周氏环的全貌目前看来是遥不可及的，但有一个被称为“同义环”的子环，它包含了几乎所有几何上有趣的类，但它承认一个显式的有限的可加性生成器集。PI与Felix Janda（密歇根大学）、Sam Grushevsky（石溪大学）和Dmitry Zakharov（中密歇根大学）合作，积极参与了这些发生器之间关系的研究。在接下来的工作中，她计划在计算上（根据生成器生成计算同义类的某些理想表达式的算法）和理论上（为类（如超椭圆轨迹）寻求新的同义表达式）发展同义交集理论。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物理学提出的等价的数学证明和扩展。例如，在与Felix Janda和Yongbin阮永斌（密歇根大学）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壁公式，这种关系与著名的物理现象——镜像对称密切相关。在未来的工作中，她将把这个过墙公式扩展到新的情况下，并用它来攻击另一个物理猜想：朗道-金兹堡/卡拉比-丘对应。她还将与alexander Buryak（利兹大学）和Ran Tessler（苏黎世联邦理工学院）合作，对有边界曲线的r-自旋理论进行理论发展，最终目标是将Witten的r-自旋猜想推广到这种情况下。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.