Tautological Intersection Theory on Moduli Spaces

PI的研究领域是代数几何。更具体地说，他研究交叉理论的几何定义类的各种模空间有关的曲线。对于这一建议，大致有两个主要的研究领域。开放轨道Gromov-Witten理论中的一系列问题来自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继续沿着PI与Hannah Markwig（萨尔布吕肯），Paul约翰逊（帝国理工学院）和Steffen Marcus（布朗大学）和Johnatan Wise（斯坦福大学）之前的工作路线。复曲面Orbifold的开放Orbifold GW理论由Brini和PI定义，其范围是对镜像对称预测的不变量给出数学定义，并获得研究（普通）GW不变量的组合技术。与罗斯的工作给出了一些积极的证据，开放不变量可能是有用的工具，如Crepant决议猜想阮和其他问题。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硬Lefschetz情况之外的Crepant解决方案猜想问题。研究了Orbifold拓扑顶点的代数结构，它与唐纳森托马斯理论中类似对象的关系，以及它与群的圈积表示理论的联系。应用技术的OGW的一个猜想的Bouchard-Klemm-Marino-Pasquetti有关这种不变量的数量出现在镜像对称通过拓扑递归由Eynard Orantin。在赫尔维茨理论的PI已调查与约翰逊和Markwig的分段多项式的双赫尔维茨号码。这三位作者对组合现象给出了令人信服且相当详尽的描述，包括一些有趣的跨壁公式。这开启了寻找这种壁交叉的几何解释的探索。作者正在寻求开发一个上同调相交公式（沿着简单Hurwitz数的ELSV公式的路线）在一些适当的模空间，描述双Hurwitz数和解释分段多项式的双有理修改的模空间，或边界修正的上同调类。一个应该证明对这个范围非常有帮助的成分是用重言式环的标准生成元来描述相对稳定映射到射影线的模空间的虚基本类到M_g-bar的推进。PI与Marcus和Wise一起在属1中研究了这个问题。理查德海恩最近提供了一个答案，所有属，但限于紧凑型曲线。PI建议在曲线的全模空间上研究这类，并利用其多项式（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合作，重新解释这个问题的热带，双重目标是获得更好的组合工具来回答这个问题，并帮助发展任意属中曲线的热带模空间的基础。 PI的研究探索了数学和数学物理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”往往是促进科学进步的一个有用的方法。拟议的研究是插入在一个肥沃的和现代的数学领域。PI参加了几次研讨会和会议。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和国际上共同组织研究学校进行了一些初步接触。他打算给一个机会，他自己的研究生前往会议，并与更广泛的数学界。最后，他有兴趣在科罗拉多州立大学和几个国际机构，包括哥斯达黎加大学和米霍坎大学之间发展伙伴关系计划。

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.