Dilogarithms

Two noteworthy events occured at King's College last Friday. One of which involved "one-hundred drunk LSE students" rioting in the lower levels of our Strand campus and the other did not. The rioters flooded our building and caused havoc, some attempted to invade the maths classrooms no doubt eager to benefit from the wisdom of our faculty, and others began ripping ceiling tiles from the English department in an attempt to seemingly tear a building to bits whilst still inside it, causing £30,000 worth of damage. The second event of note last Friday was our mathematics department colloquium which was given by the impressive Don Zagier under the title "Dilogarithms and the Bloch group: from algebraic K-theory to modular forms to conformal field theory". The rioters caused our colloquium to start late, a fact that was noted by Professor Zagier when nearly out of time he suggested that perhaps he himself could riot in order to circumvent the need to vacate the lecture thatre at the prescribed time so he could get to the end of his talk.

Permit me to pass on some brief comments from the beginning of Zagier's talk about the dilogarithm with the caveat that most of this remains beyond my ken. Nevertheless the dilogarithm has been studied by some truly great mathematicians including Euler, Leibnitz, Abel, Lobachevsky, Borel, Ramanujan, and others so if only for historical interest it is nice to look at some of the properties of the dilogarithm in passing.

We are all on first name terms with the logarithm function (that's log to you and me):
But perhaps not so well known is it's relation the dilogarithm: There are plenty of others, which are specific values, s, of the polylogarithm, or Jonquiére's function, Zagier told us that there are only 8 closed forms for the dilogarithm, some of which include the golden ratio, The only irrationals on the left hand side are powers of the golden ratio. We might look for properties of the dilogarithm that are similar to those we know of the logarithm, for example we know that, Zagier told us that there was no simple equivalent of this for the dilogarithm, but that we did have a formula, named for Abel even though it had been discovered by others before him (Spence, 1809) and after him (Lobachevsky), Zagier told those of us taking notes in the audience to just write "junk" instead of the right-hand-side of the above, however, he stressed that the order of the left-hand-side terms was important. I think practitioners of this art tend to normalise their dilogarithm functions into the Rogers L-function, which gets rid of the junk. There are a number of useful identities for the dilogarithm two of which are,These lead to equivalent forms of the five-term equation via X ~ 1/X ~ 1-X ~ 1/(1-X) ~ 1-(1/X) ~ X/(X-1). Furthermore if you define a junk-removing functional equation: Then this equation has a modulo 5 symmetry,Try it and see!

Zagier went on to look at the Bloch-Wigner dilogarithm, whose five-term equation is equal to zero and to give an argument in terms of the volume of tetrahedra in a hyperbolic 3-space modulo PSL_2(C) about why it should be zero. It sounded very nice, but I can't say I understood. For those interested the argument is given in section four of sixth set of notes by Matilde Lalin from classes given by Fernando Rodriguez Villegas called "Topics in K-Theory and L-functions". Those interested in seeing many more five-term functional equations should look through Kirillov's Dilogarithm Identities. For an introduction to dilogarithms Maximon's article seems a good place to commence.

Zagier talked fluently, rapidly and energetically about many more topics, such as the Bloch group (if you google Bloch group you may not find much helpful information but you will come across Dr. Bloch's group page), the five-term equation related to the Schrodinger equation for a cubic potential, K-theory (in particular K_3) and also rational conformal field theory, where he spoke with great enthusiasm of the recent conjectures of Werner Nahm. He never once, to my knowledge, resorted to rioting to extend his talk.