Arithmetic and Geometry of Irregular Singular Point Connections

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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.