Lecture: Strata in Complex Analysis

Leonard Max Adleman

Abstract:

What would happen if a bunch of computer scientists who knew (and perhaps still know) next to nothing about complex analysis studied it for a decade? This is not a talk about algorithms or complexity theory; it is pure math, and this is the first presentation of our results. We have looked at Riemann surfaces and cut them into flat sheets. This is nothing new, but perhaps how we have cut them and what we have done with the resulting sheets is. In particular, we have proven that the sheets, which we call strata, form a field analogous to the field of algebraic numbers and that they obey a version of the famous Kronecker-Weber theorem that all Abelian extensions of the rationals are subfields of the cyclotomic fields. Further, we have generalized the notion of power series, and used it to prove an extension of Cauchy’s famous residue theorem from integrating around poles to integrating around branch-points. We hope to use our approach to revisit Kronecker’s Jugendtraum (Hilbert’s 12th problem).

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