Arithmetic and Geometry of Irregular Singular Point Connections

本文将研究不规则连接的周期和与不规则连接相关的Gauss-Manin连接的Reimann Roch型问题。 重点将是证明高斯-马宁连接的行列式的一个简化公式。 他将研究这个公式的应用，以计算和理论的co-adjoint轨道的代表性理论和辛几何的因子。 与此同时，提议者将研究不规则连接的周期是否可以归入奇异簇的逆变动机理论。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“周期”。 从本质上讲，它们出现在几何与数论相遇的地方。 其他这样的数字（不规则周期）似乎从根本上是非几何和非数论的性质。这一建议是为了更好地理解这些不规则的时期。其关键思想是日本数学家寺真的观察到，即使人们没有这些不规则周期的几何结构，但在某些情况下，它们满足的公式与数论代数几何中出现的其他对象（称为周期因子）所满足的公式类似。

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.