Project IV (MATH4072) 2021-2022

Polylogarithms

Herbert Gangl

Description

Very similar to the well-known logarithm function, polylogarithms are special functions defined by a very simple power series. They were mentioned for the first time in 1696 in correspondence of Leibniz with Jacob Bernoulli, but then had been mostly forgotten. They have resurfaced in the last four decades in an amazing variety of areas like number theory, hyperbolic geometry, and algebraic K-theory. Moreover, and perhaps even more surprisingly, they figure prominently in Feynman integral expansions in perturbative quantum field theory, and more recently also in connection with the socalled amplituhedron and cluster algebras.

Polylogarithms and their multiple cousins satisfy lots of functional equations, and for that reason should be viewed as rather algebraic objects, despite their analytic look. Moreover, they provide higher regulator functions in algebraic number theory and also serve as volume functions in hyperbolic spaces (of odd dimension). and they should play a key role in unraveling the mystery of (mixed Tate) motives, a unifying--and still mostly elusive--theory for arithmetic properties for algebraic varieties.

Prerequisites and suggestions

Algebra II, Number Theory III or/and background in hyperbolic geometry are recommended, as there are plenty of connections to appreciate. Experience with computer algebra packages should be very valuable since there are free computer algebra packages available, for both symbolic manipulations and for numerical evaluation of these functions. They should provide a great playground for experiments and new examples for the theory without the need of too much background reading.
    A few possible directions:
  • Students with background in number theory could study `higher units' in algebraic number fields (in the so-called Bloch group).
  • There are intriguing algebraic properties pertaining to these functions, and in particular an important `fingerprint' given by the coproduct on (a formal version of) multiple polylogarithms.
  • A very timely topic would be to understand the connection of polylogarithms to cluster algebras, and also to amplitude calculations.
  • There is not so much known about finite polylogarithms which have surprising connections to the entropy function and also to classical investigations concerning Fermat's Last Theorem.

Resources

A very nice introductory article (mostly on the dilogarithm case) is the following

D. Zagier, The remarkable dilogarithm, J. Math. Phys. Sci. 22, 131-145 (1988).

A more detailed survey article is the following

D. Zagier, H. Gangl, Classical and elliptic polylogarithms and special values of L-series

A previous student gave a nice account of certain aspects of the story

J. Rhodes Polylogarithms

A somewhat advanced textbook has been written by

J Zhao Multiple zeta functions, multiple polylogarithms, and their special values

Here is a detailed pdf of a very recent talk in the `Seminaire Bourbaki' reports (Apr 2021, in French) on recent progress on a central conjecture of the theory.

C. Dupont Progres recents sur la conjecture de Zagier et le programme de Goncharov

A standard reference with lots of beautiful formulas, but not so many conceptual ideas, is

Lewin, L. (1981). Polylogarithms and Associated Functions. North-Holland-New York.

email: Herbert Gangl

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