INFO ON MATH 519: Numbers vs Functions

The analogies between numbers and functions played an important role in the development of number theory in the 20th century. The classical analogy is between integers and functions on Riemann surfaces; this analogy explicitly appears in the works of Dedekind, Hilbert, Artin, Weil, Iwasawa, Tate, Grothendieck, Mazur, Manin, Parshin and many others; cf.section 1.1 in Notes 1 . More recently other analogies were proposed; cf. this survey by Yu. I. Manin and my notes Notes 1 and Notes 2 . For a detailed exposition of the material in these notes see my books "Arithmetic Differential Equations" and "Foundations of Arithmetic Differential Geometry" (to appear, AMS 2017). The aim of the course is to present both the classical and the more modern aspects of these developments. The only prerequisites are a graduate introduction to Algebra and a graduate introduction to Complex Analysis. Some exposure to Differential Geometry would help but is not essential.

Grading is based upon attendance.

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