PECASE: Intersection Theory On Moduli Spaces

提案题目：PECASE: Moduli space的交集理论机构：Stanford university提案ID: 0238532复杂的几何物体往往有大量微妙的结构。在某种意义上，这些对象的“模空间”在一个漂亮的包中捕获了这个结构。模空间的性质是关于所讨论对象的“普遍事实”。模空间背后的思想相当古老，可以追溯到19世纪（至少）。在过去的三十年里，我们已经学会了一种研究模空间的强大方法，这要归功于格罗滕迪克学派的见解。过去十年开辟了理解这些空间的强大新方法。令人惊讶的是，动力往往来自其他领域，如理论物理或组合学。本提案旨在利用代数几何的技术来解决许多紧迫的问题，并受到其他领域的启发。这些结果反过来应该在其他领域有很强的应用。研究者还试图吸引有才华的高中生和大学生进入数学科学领域，让他们接触到令人兴奋的、先进的但又容易理解的想法，例如通过解决问题；这将主要通过斯坦福大学数学营、斯坦福大学解决问题的研讨会、伯克利数学圈和各种著作来完成。特别是，其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，研究者将在斯坦福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，邀请访客，赞助研讨会和会议，通常与其他机构联合。第三，研究者将继续通过说明文将复杂的数学思想（各个层次的）带给更广泛的读者。研究者是一个代数几何学者，他的主要兴趣是模空间上的交理论。研究者的目标是利用来自其他领域和现代机械的见解来接近几何和相关领域的许多开放和经典问题。研究者建议通过开展两个长期项目来拓宽和深化他的研究，处理数学中两个最重要的模空间：曲线的模空间，以及格拉斯曼及其推广。第一个项目将使用现代技术来阐明曲线模空间的上同（或Chow）环（“同义环”）的“几何自然”部分（“同义环”）背后的推测和已知组合结构。第二个项目将使用代数几何思想来解决关于Littlewood-Richardson规则、Schubert问题和推广到其他群体背后的结构（代数、算术、几何、枚举等）的经典开放问题。第一个项目涉及物理、拓扑学、组合学、可积系统和辛几何；第二种涉及组合学、表示理论和算术几何。因此，两者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的工作关系和合作。该项目最初是作为职业奖资助的，并于2004年9月转换为工程师和科学家的总统早期职业奖（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.