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Arakelov theory and quantum cohomology of homogeneous varieties

阿拉克洛夫理论和齐次簇的量子上同调

基本信息

批准号：
0401082
负责人：
Harry Tamvakis
金额：
$ 9.7万
依托单位：
Brandeis University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2004
资助国家：
美国
起止时间：
2004-09-01 至 2006-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401082&HistoricalAwards=false
关键词：
Arakelov theory quantum cohomology homogeneous

项目摘要

The investigator will study two different intersection theories inalgebraic geometry: the arithmetic Chow rings of varieties definedover the integers and the quantum cohomology rings of complexmanifolds. The varieties considered are homogeneous spaces ofLie groups, and the goal is to understand the arithmetic and quantumSchubert calculus. This allows one to effectively compute numericalinvariants such as heights, on the one hand, and Gromov-Witteninvariants, on the other. A further aim is to answer related questionsabout the combinatorics of modern Schubert calculus and thepolynomials which represent degeneracy loci of vector bundles.The study of the symmetries observed in geometric objects has been ameeting point of mathematics and physics for a long time. The manyexperiments and computations made in the 19th century eventually ledto the cohomology and intersection theories of 20th century mathematics,which were applied to solve important open problems in both fields. Atpresent, we face a similar situation in modern number theory and stringtheory, and computations of concrete examples are crucial to guide ourintuition and to better understand the general theories. The investigatorstudies two new intersection theories, Arakelov theory and quantumcohomology, in many examples which are good testing grounds forboth of them. The potential applications are a better understanding ofDiophantine equations and approximation, used in coding theory andtheoretical computer science, and the enumerative geometry of curves,related to classical algebraic geometry and quantum field theory in physics.

研究者将研究代数几何中两种不同的交理论：整数上定义的簇的算术Chow环和复流形的量子上同调环。考虑的品种是齐性空间的李群，目标是了解算术和quantumSchubert微积分。这使得一个有效地计算numericalinvariants，如高度，一方面，和Gromov-Witteninvariants，另一方面。进一步的目的是回答现代Schubert微积分的组合学和表示向量丛退化轨迹的多项式的有关问题。世纪的许多实验和计算最终导致了20世纪世纪数学的上同调和交理论，这些理论被应用于解决这两个领域的重要开放问题。目前，我们在现代数论和弦论中面临着类似的情况，具体例子的计算对于引导我们的直觉和更好地理解一般理论至关重要。本文研究了两个新的相交理论，Arakelov理论和量子上同调，并给出了大量的例子，这些例子是检验这两个理论的良好基础。潜在的应用是更好地理解丢番图方程和近似，用于编码理论和理论计算机科学，以及曲线的枚举几何，与经典代数几何和物理学中的量子场论有关。