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形状。更具体地说，他研究了与曲线相关的各种模量空间上的几何定义类的交点理论。对于此提案，大约有两个主要的研究领域。 PI先前与安德里亚·布里尼（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）（Colorado State University）与Andrea Brini（Colorado State University）的著作中的一系列问题。 Hurwitz理论中的第二组问题继续沿PI与Hannah Markwig（Saarbrucken），Paul Johnson（帝国学院）和Steffen Marcus（布朗大学）和约翰坦·怀斯（Stanford Wise）（斯坦福大学）（斯坦福大学）的著作。 Brini和Pi定义了针对曲折的开放式Orbifold GW理论，其范围的范围是对镜子对称性预测的不变性定义，并获得Orbifolds的GW不变性的组合技术。与罗斯的作品提供了一些积极的证据，表明公开不变的人可能是诸如Ruan和其他人的Crepant解决方案的问题。我们提到了该领域未来调查的一些方向： - 为开放的GW不变式设置方便的形式主义，这可能有助于将这些工具的适用性扩展到毛茸茸的解决方案的猜想问题之外，超越了硬质量的案例。 OGW的技术对Bouchard-klemm-marino-pasquetti的猜想，将这种不变性与通过Eynard-orortin.in Hurwitz Theory开发的拓扑递归在镜像对称性中产生的数量有关，PI与Johnson和Markwig the Markwig the Markwig the Markwig the Markwig the Markwig the Markwig the零件hurwitz数字。这三位作者对组合现象进行了令人信服且相当详尽的描述，其中包括一些有趣的墙壁交叉公式。这打开了寻找这种墙壁交叉点的几何解释的追求。作者正在寻求在一些适当的模量空间上开发共同体的交集公式（沿着Elsv公式的线条，沿着Elsv公式的线条，用于简单的Hurwitz数字），该空间描述了双Hurwitz数字，并在涉及的Moduli空间或涉及的边界校正的涉及的型号修饰的范围内解释了分段多项式。应该证明对此范围非常有用的成分是将推送到m_g-bar的描述，该曲线是根据重言式环的标准生成器而言，相对稳定地图的虚拟基本模数空间相对稳定地图与投影线相对稳定。 PI与Marcus一起研究了这个问题。RichardHain最近为所有属但限制了紧凑型曲线的答案。 PI提议在曲线的完整模量空间上研究此类，并利用其多项式（或分段多项式）属性，以证明具有双hurwitz数字的Elsv型公式。 PI还打算与Hannah Markwig合作，以热区重新诠释这个问题，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以任意属的曲线的热带模量空间的基础发展。 PI的研究探讨了数学和数学物理学几个领域之间的互连。在几个科学学科中创建“桥梁”和“词典”通常是使科学进步的有用方法。拟议的研究插入了数学的肥沃和现代领域。 PI已参加了几次研讨会和会议。在拟议项目的整个期间，PI打算支持研究生并帮助他们的数学发展。他对科罗拉多州和国际上的研究学校进行了一些初步联系。他打算为自己的研究生提供机会参加会议，并与更广泛的数学社区互动。最终，他有兴趣在科罗拉多州立大学和包括哥斯达黎加大学和米歇纳大学在内的几个国际机构之间制定合作计划。