CAREER: The cohomology and birational geometry of moduli spaces

职业：模空间的上同调和双有理几何

• 批准号: 0952535
• 负责人: Izzet Coskun
• 金额: $ 40万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0952535&HistoricalAwards=false
• 关键词: CAREER; cohomology; birational; geometry; moduli

Homogeneous varieties, in particular, ordinary and isotropic Grassmannians, are central objects of study in algebraic geometry, representation theory, and combinatorics. The investigator proposes to develop positive algorithms for computing the structure constants of the cohomology of isotropic Grassmannians and isotropic flag varieties. In recent years, similar positive algorithms have led to the solutions of many important problems for Type A Grassmannians, including the saturation conjecture and the reality of Schubert calculus. The investigator will also study the effective cones of the moduli space of curves and stable maps. The moduli spaces of curves are among the most studied objects in mathematics. Their cones of effective divisors are important invariants, intimately tied to problems such as the Schottky problem, the Kodaira dimension of the moduli space of curves, and the existence of modular forms. Recently, understanding the effective cone of moduli spaces has received new impetus following the seminal work of Hacon, McKernan, and their collaborators on the Minimal Model Program (MMP). The investigator proposes to run MMP on moduli spaces such as the Kontsevich moduli spaces of genus zero stable maps or Hilbert scheme of points on the projective plane. Systems of polynomial equations occur in many facets of life ranging from evolutionary biology to physics and from cryptography to computer science. Algebraic geometry studies geometric properties of solutions of polynomial systems. The solutions that have many symmetries are especially interesting and important. For example, a sphere is a perfectly symmetric space in the sense that any point can be rotated to any other point. The investigator studies the number of solutions to polynomial systems involving such perfectly symmetric spaces called homogeneous varieties. The investigator also calculates more subtle geometric invariants of systems of polynomial equations such as how the behavior of the space of solutions changes under perturbations of the system.

齐次簇，特别是普通格拉斯曼函数和各向同性格拉斯曼函数，是代数几何、表示论和组合学的中心研究对象。研究人员建议开发正算法来计算各向同性格拉斯曼函数和各向同性标志簇的上同调的结构常数。近年来，类似的正算法导致了A型格拉斯曼主义者许多重要问题的解决，包括饱和猜想和舒伯特微积分的现实。研究人员还将研究曲线和稳定图模空间的有效锥体。曲线的模空间是数学中研究最多的对象之一。它们的有效除数锥体是重要的不变量，与肖特基问题、曲线模空间的 Kodaira 维数以及模形式的存在等问题密切相关。最近，继 Hacon、McKernan 及其合作者在最小模型程序 (MMP) 方面的开创性工作之后，对模空间有效锥的理解获得了新的推动力。研究人员建议在模空间上运行 MMP，例如属零稳定映射的 Kontsevich 模空间或射影平面上点的希尔伯特方案。多项式方程组出现在生活的许多方面，从进化生物学到物理学，从密码学到​​计算机科学。代数几何研究多项式系统解的几何性质。 具有许多对称性的解特别有趣且重要。例如，球体是一个完全对称的空间，因为任何点都可以旋转到任何其他点。研究人员研究涉及称为齐次簇的完全对称空间的多项式系统的解的数量。研究人员还计算多项式方程组的更微妙的几何不变量，例如解空间的行为在系统扰动下如何变化。