Tautological Intersection Theory on Moduli Spaces

模空间的同义反复交集理论

• 批准号: 1101549
• 负责人: Renzo Cavalieri
• 金额: $ 11.34万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101549&HistoricalAwards=false
• 关键词: Tautological; Intersection; Theory; Moduli; Spaces

The PI's field of research is algebraic geometry. More specifically he investigates the intersection theory of geometrically defined classes on various moduli spaces related to curves. For this proposal there are roughly two major areas of research. A cluster of questions in open orbifold Gromov-Witten theory spring from the PI's previous work with Andrea Brini (University of Geneva) and his own graduate student Dusty Ross (Colorado State University). A second group of problems in Hurwitz theory continue along the line of the PI's previous work with Hannah Markwig (Saarbrucken), Paul Johnson (Imperial College) and Steffen Marcus (Brown University) and Johnatan Wise (Stanford University). Open orbifold GW theory for toric orbifolds was defined by Brini and the PI, with the scope of giving a mathematical definition to invariants predicted by mirror symmetry and to obtain combinatorial techniques to study (ordinary) GW invariants of orbifolds. The work with Ross gave some positive evidence that open invariants could be useful tool for questions such as the Crepant Resolution Conjectures of Ruan and others. We mention some direction for future investigation in the area:- setting up a convenient formalism for open GW invariants, that may help extend the applicability of these tools to the Crepant Resolution Conjecture question beyond the Hard Lefschetz cases.- studying the algebraic structure of the orbifold topological vertex, its relation with the analogous object in Donaldson Thomas theory, and its connection with representation theory of wreath products of groups.- applying techniques of OGW to a conjecture of Bouchard-Klemm-Marino-Pasquetti relating such invariants to quantities arising in mirror symmetry via topological recursions developed by Eynard-Orantin.In Hurwitz theory the PI has investigated with Johnson and Markwig the piecewise polynomiality of double Hurwitz numbers. The three authors gave a convincing and fairly exhaustive description of the combinatorial phenomenon, including some interesting wall crossing formulas. This opens up the quest of finding a geometric interpretation for such wall crossings. The author is seeking to develop a cohomological intersection formula (along the lines of the ELSV formula for simple Hurwitz numbers) on some appropriate moduli spaces that describes the double Hurwitz numbers and explains the piecewise polynomiality either in terms of birational modification of the moduli spaces involved, or of boundary corrections to the cohomology classes involved. An ingredient that should prove extremely helpful to this scope is the description of the pushforward to M_g-bar of the virtual fundamental class of moduli spaces of relative stable maps to the projective line in terms of standard generators of the tautological ring. This question has been investigated by the PI together with Marcus and Wise in genus 1. Richard Hain recently provides an answer for all genera but restricting to curves of compact type. The PI proposes to investigate this class on the full moduli space of curves, and exploit its polynomiality (or piecewise polynomiality) properties to prove an ELSV-type formula for double Hurwitz numbers. The PI also intends to work with Hannah Markwig to reinterpret this question tropically, with the twofold goal of obtaining better combinatorial tools to answer the question and to help the developments of the foundations of tropical moduli spaces of curves in arbitrary genus. The PI's research explores interconnections among several areas of mathematics and mathematical physics. Creating "bridges" and "dictionaries" among several scientific disciplines is often a useful way to make science progress. The proposed research is inserted in a fertile and modern area of mathematics. The PI has been participating to several workshops and conferences. Throughout the period of the proposed project, the PI intends to support graduate students and help their mathematical development. He has taken some preliminary contacts about co-organizing research schools both in Colorado and internationally. He intends to give an opportunity to his own graduate students to travel to conferences and to interact with the broader mathematical community. Finally he is interested in developing partnership programs between Colorado State University and several international institutions, including University of Costa Rica, and University of Michoacan.

PI的研究领域是代数几何。更具体地说，他研究了与曲线相关的各种模空间上几何定义类的交集理论。对于该提案，大致有两个主要研究领域。开环格罗莫夫-维滕理论中的一系列问题源于 PI 之前与 Andrea Brini（日内瓦大学）和他自己的研究生 Dusty Ross（科罗拉多州立大学）的合作。赫尔维茨理论中的第二组问题沿着 PI 之前与汉娜·马克维格（萨尔布吕肯）、保罗·约翰逊（帝国理工学院）、斯特芬·马库斯（布朗大学）和乔纳坦·怀斯（斯坦福大学）的工作路线继续解决。 Brini 和 PI 定义了环面 Orbifold 的开放 Orbifold GW 理论，其范围是对镜像对称预测的不变量给出数学定义，并获得组合技术来研究 Orbifolds 的（普通）GW 不变量。与 Ross 的合作提供了一些积极的证据，表明开不变量可能是解决诸如 Ruan 等人的 Crepant 消解猜想等问题的有用工具。我们提到了该领域未来研究的一些方向： - 为开放 GW 不变量建立一个方便的形式主义，这可能有助于将这些工具的适用性扩展到 Hard Lefschetz 案例之外的 Crepant 解析猜想问题。 - 研究轨道拓扑顶点，它与唐纳森托马斯理论中的类似对象的关系，以及它与群的花环积表示论的联系。-将OGW技术应用于猜想Bouchard-Klemm-Marino-Pasquetti 通过 Eynard-Orantin 开发的拓扑递归将这些不变量与镜像对称中产生的量联系起来。 在 Hurwitz 理论中，PI 与 Johnson 和 Markwig 一起研究了双 Hurwitz 数的分段多项式。三位作者对组合现象给出了令人信服且相当详尽的描述，包括一些有趣的穿墙公式。这就开启了寻找此类墙壁交叉点的几何解释的探索。作者正在寻求在一些适当的模空间上开发一个上同调交集公式（沿着简单 Hurwitz 数的 ELSV 公式），该模空间描述双 Hurwitz 数并根据所涉及的模空间的双有理修改来解释分段多项式，或所涉及的上同调类的边界修正。一个对这个范围非常有帮助的成分是描述相对稳定映射的模空间的虚拟基本类 M_g-bar 到同义反复环的标准生成元的射影线的推进。 PI 与 Marcus 和 Wise 一起在属 1 中研究了这个问题。Richard Hain 最近为所有属提供了答案，但仅限于紧凑型曲线。 PI 建议在曲线的完整模空间上研究此类，并利用其多项式（或分段多项式）属性来证明双 Hurwitz 数的 ELSV 型公式。 PI 还打算与 Hannah Markwig 合作，从热带角度重新解释这个问题，其双重目标是获得更好的组合工具来回答这个问题，并帮助发展任意属曲线的热带模空间基础。 PI 的研究探索了数学和数学物理多个领域之间的相互联系。在多个科学学科之间建立“桥梁”和“词典”往往是推动科学进步的有效途径。拟议的研究被插入到一个丰富而现代的数学领域。 PI 参加了多个研讨会和会议。在整个拟议项目期间，PI 打算支持研究生并帮助他们的数学发展。他已经就在科罗拉多州和国际上共同组织研究学校进行了一些初步接触。他打算为自己的研究生提供参加会议并与更广泛的数学界互动的机会。最后，他有兴趣在科罗拉多州立大学与几个国际机构（包括哥斯达黎加大学和米却肯大学）之间建立合作伙伴关系。