Arithmetic and Geometry of Irregular Singular Point Connections

不规则奇点连接的算术和几何

• 批准号: 0103765
• 负责人: Spencer Bloch
• 金额: $ 28.02万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103765&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Irregular; Singular; Point

The proposer will study periods of irregular connections and Reimann Roch type problems for Gauss-Manin connections associated to irregular connections. The focus will be on proving a conjectured formula for the determinant of the Gauss-Manin connection. He will study applications of this formula to epsilon factors in arithmetic and to the theory of co-adjoint orbits in representation theory and symplectic geometry. At the same time, the proposer will investigate whether periods of irregular connections can be subsumed in a theory of contravariant motives for singular varieties.Certain numbers like pi and e play a central role in mathematics. In some cases, these numbers are "periods". Essentially, they arise where geometry meets number theory. Other such numbers (irregular periods) seem to be fundamentally non-geometric and non-number theoretic in nature. This proposal is an attempt to better understand these irregular periods. The key idea is the observation of a Japanese mathematician, Terasoma, that even though one has no geometric construction of these irregular periods, they are known in some cases to satisfy formulae which are analogous to the formulae satisfied by other objects (called epsilon factors) arising in number theoretic algebraic geometry.

本文将研究不规则连接的周期和与不规则连接相关的高斯-马宁连接的Reimann - Roch型问题。重点将是证明高斯-马宁联系的行列式的一个猜想公式。他将研究这个公式在算术中的epsilon因子以及在表示理论和辛几何中的协伴随轨道理论中的应用。同时，提议者将研究不规则连接的周期是否可以包含在奇异变量的逆变动机理论中。像圆周率和e这样的数字在数学中起着核心作用。在某些情况下，这些数字是“句号”。从本质上讲，它们出现在几何和数论相遇的地方。其他这样的数字（不规则周期）在本质上似乎从根本上是非几何和非数论的。这个建议是为了更好地理解这些不规则周期。关键思想是日本数学家Terasoma的观察，他认为，即使没有这些不规则周期的几何结构，在某些情况下，它们满足的公式与数论代数几何中出现的其他对象（称为epsilon因子）所满足的公式类似。