Irregular connections and the Stokes phenomena

Abstract: Deligne's book focuses mostly on connections with regular singularities -- in the 1970s, Deligne found connections with irregular singularities to be pathological (see his article "Pourquoi un géomètre algébriste s'intéresse-t-il aux connexions irrégulières?"). But since then, Deligne, Malgrange, Sibuya, and many others have noticed that irregular connections are home to many interesting phenomena which seem to mirror things occurring for ell-adic sheaves.

Regular connections are the simplest to understand since, by Riemann-Hilbert, they are completely determined by the monodromy of their solutions. Unfortunately, this fails for irregular connections -- there are nontrivial irregular connections whose solutions have no monodromy. In this talk we describe the Stokes data which one can use to help understand irregular connections.

Reference: Malgrange, Équations Différentielles à Coefficients Polynomiaux, chapters 3 and 4 Babbitt and Varadarajan, Local moduli for meromorphic differential equations Deligne, Malgrange, and Ramis, Singularités Irrégulières: Correspondance et documents, particularly the 19.4.78 letter from Deligne to Malgrange.

algebraic geometrynumber theory

Audience: advanced learners

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2025 topic: Weil conjectures.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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