Irregularity of algebraic differential equations on varieties in positive characteristic

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This project proposal is concerned with systems of algebraic differential equations on smooth varieties X over a field of positive characteristic. Such systems are also called "stratified bundles''. The corresponding objects on complex manifolds are vector bundles with flat connection, but in positive characteristic new phenomena appear. In particular, the boundary behavior of a stratified bundle bears close resemblance to the ramification theory of l-adic local systems.The overall goal of this project is to further develop this analogy. In 2012, P. Deligne proved a finiteness theorem for irreducible l-adic local systems of fixed rank, with ramification bounded by a fixed divisor supported at infinity. The concrete goal of this proposal is to define and study the notion of a "stratified bundle with irregularity bounded by a divisor supported at infinity''. One successful outcome of the project would be to be able to formulate a statement about stratified bundles, which is analogous to Deligne's theorem.To achieve this, I first intend to assume that X is a curve and to study the structure of a stratified bundle formally locally around a point at infinity. Here I plan to construct an invariant measuring the quality of its singularity, which should be analogous to the irregularity number of a flat connection and the Swan conductor of an l-adic local system. If X has higher dimension, I plan to define the notion of bounded irregularity by probing X with curves.