Project IV, 2017-2018

Continued fractions

Anna Felikson and Pavel Tumarkin

Description:

A continued fraction is an expression of the form

[a_{0}; a_{1}, a_{2}, a_{3}, \dots] = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{\dots}}}}

Continued fractions have been studied since at least 1572 when R. Bombelli, and later P. Cataldi, used them to obtain approximations to √13 and √18, respectively. Since then continued fractions found their numerous connections and applications in

number theory,
geometry of lattices,
hyperbolic geometry,
combinatorics,
knots and links,
cluster algebras,
and many other branches of mathematics.

In particular, continued fractions provide relations between different domains of mathematics.

Anna Felikson will supervise in Michaelmas and Pavel Tumarkin in Epiphany.

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Prerequisites and corequisites: Algebra II is a prerequisite. Geometry III/IV would be useful as a corequisite.

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Resources:

One can take a look at Wikipedia page on Continued fractions.
An accesible introduction to number-theoretic and measure-theoretic aspects of continued fractions Continued fractions by A.Ya. Khinchin.
Some geometric aspects of the story can be found in Continued fractions and hyperbolic geometry by Caroline Series.
A book on continued fractions and their multidimensional analogues Geometry of Continued fractions by O. Karpenkov ( library link).
And some open problems: Open problems in Geometry of Continued fractions by O. Karpenkov.

More reading on continued frections can be found here.

emails: Anna Felikson, Pavel Tumarkin

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