PECASE: Intersection Theory On Moduli Spaces

提案标题：PECASE：模空间的交集理论机构：斯坦福大学提案ID：0238532复杂的几何对象通常具有大量的微妙结构。在某种意义上，这些物体的“模空间”在一个很好的包装中捕捉到了这种结构。模空间的性质是关于所讨论对象的“普遍事实”。模空间背后的思想是相当古老的，至少可以追溯到19世纪。在过去的三十年里，由于Grothendieck学派的深刻见解，我们已经学会了一种研究模空间的强有力的方法。过去的十年为理解这些空间开辟了强有力的新途径。令人惊讶的是，推动力往往来自其他领域，如理论物理或组合学。这项建议寻求使用代数几何中的技术来解决许多紧迫的问题，并受到其他领域见解的高度推动。反过来，这些成果也应该在其他领域有很强的应用价值。研究人员还试图吸引有才华的高中生和本科生进入数学科学领域，让他们接触到令人兴奋和先进但容易理解的想法，例如通过解决问题；这将主要通过斯坦福大学数学夏令营、斯坦福大学的问题解决研讨会、伯克利数学圈和各种写作来完成。特别是，目标是从以前未开发的人才库中吸引学生。其次，在研究生层面，研究人员将在斯坦福大学建立一个代数几何中心，为研究生和博士后提供资源，开发新课程，邀请参观者，并赞助研讨会和会议，通常是与其他机构合作。第三，调查员将继续通过说明性写作将(所有级别的)复杂的数学思想带给更广泛的受众。这位研究者是一位代数几何学家，他的主要兴趣是模空间上的交理论。研究人员的目标是利用其他领域和现代机械的见解来解决几何学和相关领域中的许多开放的和经典的问题。这位研究人员建议通过开展两个较长期的项目来拓宽和深化他的研究，该项目涉及数学中最重要的两个模空间：曲线的模空间和Grassmanian及其推广。第一个项目将使用现代技术来阐明曲线的模空间的上同调环(或Chow环)的“几何自然”部分(“重言环”)背后的猜想和已知的组合结构。第二个项目将使用代数几何的思想来解决关于Littlewood-Richardson规则、Schubert问题以及对其他群的推广背后的结构(代数、算术、几何、计数等)的经典开放问题。第一个项目涉及物理学、拓扑学、组合学、可积系统和辛几何；第二个项目涉及组合学、表示论和算术几何。因此，这两个项目都将涉及在广泛的不同领域建立知识基础，以及与这些领域的研究人员建立终身工作关系和合作。该项目最初是作为职业奖资助的，并于2004年9月转变为总统工程师和科学家早期职业奖(PECASE)奖。

Proposal Title: PECASE: Intersection Theory On Moduli SpacesInstitution: Stanford UniversityProposal ID: 0238532Complicated geometric objects often have a great deal of subtle structure. ``Moduli spaces'' for these objects in some sense capture this structure in a nice package. Properties of moduli spaces are ``universal facts'' about the objects in question. Ideas behind moduli spaces are quite old, dating back to the nineteenth century (at least). In the last thirty years, we have learned a powerful way of studying moduli spaces, thanks to the insights of Grothendieck's school. The last decade has opened up powerful new ways of understanding these spaces. Surprisingly, the impetus often came from other fields, such as theoretical physics or combinatorics. This proposal seeks to approach many pressing problems using techniques from algebraic geometry, highly motivated by insights from other fields. The results in turn should have strong applications in other fields. The investigator also seeks to attract talented high school and undergraduate students into the mathematical sciences, by exposing them to exciting and advanced yet accessible ideas, for example through problem solving; this will be done primarily through the Stanford University Math Camp, a problem solving seminar at Stanford, the Berkeley Math Circle, and various writings. In particular, the goal is to attract students from previously untapped pools of talent. Second, at the graduate level, the investigator will build a center for algebraic geometry at Stanford, by providing resources for graduate students and postdoctoral students, developing new courses, inviting visitors, and sponsoring seminars and conferences, often jointly with other institutions. Third, the investigator will continue to bring sophisticated mathematical ideas (of all levels) to a wider audience through expository writing. The investigator is an algebraic geometer whose primary interest is in intersection theory on moduli spaces. The investigator's goal is to approach many open and classical questions in geometry and related fields using both insights from other fields and modern machinery. The investigator proposes to broaden and deepen his research, by undertaking two longer-term projects, dealing with two of the most important moduli spaces in mathematics: the moduli space of curves, and the Grassmannian and its generalizations. The first project will use modern techniques to illuminate the conjectural and known combinatorial structure behind the ``geometrically natural'' part of the cohomology (or Chow) ring of moduli space of curves (the ``tautological ring''). The second project will use algebro-geometric ideas to solve classical open questions about the structure (algebraic, arithmetic, geometric, enumerative, and more) behind Littlewood-Richardson rules, Schubert problems, and generalizations to other groups. The first project relates to physics, topology, combinatorics, integrable systems, and symplectic geometry; the second involves combinatorics, representation theory, and arithmetic geometry. Thus both will involve developing a base of knowledge in a broad array of different fields, as well as lifelong working relationships and collaborations with researchers in these fields.This project was originally funded as a CAREER award, and was converted to a Presidential Early Career Award for Engineers and Scientists (PECASE) award in September 2004.