Project IV (MATH4072) 2007-2008

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 three decades in an amazing variety of areas like number theory, hyperbolic geometry, algebraic K-theory and, perhaps even more surprisingly, in Feynman integral expansions in physics, for example in perturbative quantum field theory.

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

The Wikipedia entry for polylogarithms lists mainly rather classical formulas, disregarding the wealth of new connections to other fields---it could be part of the project to provide a more up-to-date entry, or to even write a survey to which that entry then links.

Prerequisites and suggestions

This project is a comparably demanding one in that the literature for it is rather scattered; in particular there is no basic textbook for it available. On the other hand, there will be a lot of activity in this direction, starting from scratch, during the fall term (e.g. in the arithmetic study group, several visitors--both students and researchers--are expected). Furthermore, the topic can provide a playground for (computer-aided) experiments without too much background reading, for example:
  • students with background in number theory could study higher units in algebraic number fields (in the so-called Bloch group);
  • if basic knowledge in both hyperbolic geometry and number theory can be assumed, an intriguing question could be to implement an algorithm for finding certain fundamental domains (of SL_2(O_K), K imaginary quadratic, acting on hyperbolic 3-space);
  • one could try to write a program to verify--or even find--functional equations for polylogarithms.
More combinatorially inclined students could investigate finite polylogarithms, or else study configurations of points in projective space in connection with the "geometry" of the polylogarithm.

Resources (updated)

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 more broad survey (also covering multiple polylogarithms and multiple zeta values (in French) is given in the Bourbaki talk of

P. Cartier Fonctions polylogarithmiques, nombres polyzetas et groupes pro-unipotentes

For an introductory glimpse into the calculus involved for functional equations check out

H. Gangl, Functional equations of polylogarithms.

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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