Arithmetic and Geometry of Irregular Singular Point Connections

作者将研究不规则联络的周期和与不规则联络相关的Gauss-Manin联络的Reimann Roch型问题。重点是证明高斯-马宁关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因子中的应用，以及在表象理论和辛几何中的共伴轨道理论的应用。同时，提出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。某些数字，如pi和e，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，它们是在几何与数论相遇的地方出现的。其他这样的数字(不规则周期)似乎本质上是非几何和非数论的。这项提议是为了更好地理解这些不规律的时期。其核心思想是日本数学家Terasoma的观察，即使没有这些不规则周期的几何结构，它们在某些情况下也已知满足类似于数论代数几何中出现的其他对象所满足的公式(称为epsilon因子)的公式。

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.