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;
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.
Qualified students with disabilities needing appropriate academic adjustments
should contact me as soon as possible to ensure your needs
are met in a timely manner. Handouts are available
in alternative accessible formats upon request.
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