Tautological Intersection Theory on Moduli Spaces

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。对于这一提议，大致有两个主要的研究领域。开放的奥比诺德·格罗莫夫-维滕理论中的一系列问题源于PI之前与安德里亚·布里尼(日内瓦大学)和他自己的研究生达斯蒂·罗斯(科罗拉多州立大学)的合作。赫维茨理论中的第二组问题继续沿着PI之前与Hannah Markwig(Saarbrucken)、Paul Johnson(帝国学院)、Steffen Marcus(布朗大学)和Johnatan Wise(斯坦福大学)的工作路线。由Brini和PI定义了圆环不变量的开环不变量GW理论，其范围是给出镜像对称性所预言的不变量的数学定义，并获得研究(普通)GW不变量的组合方法。与Ross的工作给出了一些积极的证据，表明开不变量可以作为解决Ruan等人的Crepant解析猜想等问题的有用工具。我们指出了这一领域未来研究的一些方向：-为开GW不变量建立一个方便的形式论，这可能有助于将这些工具的适用性扩展到复杂的Lefschetz情形之外的Crepant解析猜想问题。-研究orbiold拓扑点的代数结构，它与Donaldson Thomas理论中类似对象的关系，以及它与群的圈积表示理论的关系。-将OGW的技巧应用于Bouchard-Klemm-Marino-Pasquetti猜想，通过拓扑递归将这些不变量与出现在镜像对称中的量联系起来。在Hurwitz理论中，PI与Johnson和Markwig研究了双Hurwitz数的分段多项式性质。这三位作者对这种组合现象给出了令人信服和相当详尽的描述，包括一些有趣的穿越墙的公式。这开启了为这种跨越墙寻找几何解释的探索。作者试图在适当的模空间上建立一个上同调交公式(沿着简单Hurwitz数的ELSV公式)，它描述了双Hurwitz数，并用所涉及的模空间的双调变或所涉及的上同调类的边界修正来解释分段多项式.一个对这个范围非常有帮助的成分是用重言环的标准生成元描述相对稳定映射到投影线的虚基本模类到M_g-bar的推进。这个问题已经由PI与Marcus和Wise一起在亏格1中进行了研究。Richard Hain最近给出了所有属的答案，但仅限于紧凑型曲线。PI建议在全模空间上研究这类曲线，并利用它的多项式(或分段多项式)性质来证明双Hurwitz数的ELSV型公式。PI还打算与Hannah Markwig一起从热带重新解释这个问题，有两个目标：获得更好的组合工具来回答这个问题，并帮助发展任意亏格曲线的热带模空间的基础。国际数学联合会的研究探索了数学和数学物理的几个领域之间的相互联系。在几个科学学科之间建立“桥梁”和“词典”往往是推动科学进步的一种有用方式。这项拟议的研究被插入到一个肥沃的现代数学领域。国际和平协会参加了几个讲习班和会议。在整个拟议的项目期间，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型学校一事进行了一些初步接触。他打算给自己的研究生一个机会，让他们去参加会议，并与更广泛的数学界互动。最后，他对科罗拉多州立大学与包括哥斯达黎加大学和米却肯大学在内的几个国际机构之间的合作项目感兴趣。

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.