What do mathematicians do all day?

While in graduate school I’ve spent a while wrestling with the question,

What is the purpose of mathematics?

It’s an eminently practical obsession – no one is going to tell me best to “do mathematics”, so it’s crucial to find a perpective that will let me navigate the ocean of mathematical thought. There are many common answers to why people do mathematics – say, “beauty” (whatever that is), “applications in the far future” (but why not sooner?), “curiosity” (better, but isn’t this deeply selfish?) – and I have usually found these anwers a bit trite.

I’ve been running into Mixed Tate motives recently, which are related to all kinds of stuff, and particular, are related to the very classical polylogarithm.

Definition: The polylogarithm function is $Li_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s} = z + \frac{z^2}{2^s} + \frac{z^3}{3^s} + \ldots .$

Fixing $z=1$ one recovers the Riemann zeta $\zeta(s) = Li_s(1)$ ; fixing $s=1$ one recovers the logarithm $Li_1(z) = -\log(1-z)$ . Multiple-variable generalizations of this function satisfy remarkable identities called the shuffle relations, which have a “motivic origin” and which I cannot give any justice due to lack of expertise. However, I can tell a quick story that illustrates something funny about the way mathematics seems to work.

Specializing $Li_s(1)$ further, one recovers the Riemann zeta values $\zeta(3), \zeta(5), \ldots$ . Now, these are theonly “interesting” values of the Riemann zeta function at integers, because the values of $\zeta$ for positive even integers can be immediately calculated from the negative integer $\zeta$ -values by the functional equaiton, and at the negative integers the values of $\zeta$ are in fact rational:

$\zeta(-n) = (-1)^n \frac{B_{n+1}}{n+1}$

where $B_k$ is the $k$-th Bernoulli number, which is most easily defined by the generating function $\frac{t}{1-e^{-t}} = \sum_{m=0}^\infty B_m \frac{t^m}{m!}.$

The values of $\zeta$ at the odd positive integers are conjectured to be all transcedental, although little is known. Only in 1979 was it proven by Apery that $\zeta(3)$ is irrational. If we believe that humans are doing a breadth-first search down the tree of knowledge, then this fact is noticeably “harder” than the discovery of the cosmic microwave backgorund, which was already known by 1964.

One can give an immediate generalization of the zeta function to a multiple variable function: $\zeta(n_1, n_2, \ldots, n_m) = \sum_{0 < k_1 < \ldots < k_m} \frac{1}{k_1^{n_1}k_2^{n_2} \ldots k_m^{n_m}}.$

Now it was extremely classical that these integer values of the zeta functions are integrals of rational functions over polyhedra (!): this follows by repeatedly applying the following recursion relation for the polylogarithm

$Li_{s+1}(z) = \int_0^z \frac{Li_s(t)}{t} dt$

and specializing to $z=1$ .

However, the problem of an integral representation for the multiple zeta function was open for a very long time, until, supposedly, Don Zagier started investigating some interesting algebraic relations between multiple zeta values and got around to asking his (former?) student Kontsevich whether he had any ideas where these relations were coming from. To this apaprently Kontsevich immediately replied that for integer $n_i$ one has

$\zeta(n_1, \ldots, n_m) = \int_\Delta \frac{dt_{1,1}}{1-t_{1,1}}\frac{dt_{1,2}}{t_{1,2}}\ldots \frac{dt_{1, n_1}}{t_{1, n_1}}\frac{dt_{2,1}}{1-t_{2,1}} \frac{dt_{2,2}}{t_{2,2}} \ldots \frac{dt_{m, n_m}}{t_{n, n_m}},$ where $\Delta = \{ 0 \leq t_{1,1} \leq t_{1,2} \leq \ldots \leq t_{1, n_1} \leq t_{2, 1} \leq \ldots \leq t_{n, n_m} \leq 1.$

This follows from a recursive formula for a multiple-variable generalization of the polylogarithm, immediately leads to many identities, and eventually leads to relations between these things and quantum groups and motives and galois groups and scattering amplitudes and much, nuch more.

I’ve been told this story by hearsay, but I think this is consistent with the literature; for example, Goncharov cites Konsevich for this discovery and the citation is “personal communication” - ha! I think that there are a few things one can learn from this story:

A) You never know what you already know until you ask yourself what you know.

B) Don’t underestimate Kontsevich.

C) A central theme in mathematics is the search for surprising structure in timeless, elementary things. Sums of powers of integers are one of the simplest expressions one can write down. They’re going to pop up everywhere because they’re simple. So one should understand them deeply, and more likely than not, there’s something deep to be found.

Another special function that people use a lot is the entropy of a discrete random variable

$S = \sum_i p_i \log p_i.$

This quantity was (as far as I know) defined by Claude Shannon, as inspired by the entropy quantity of statistical thermodynamics. There are two reasonable ways of interpreting the “meaning” of this function:

information is something “physical” (whatever that means) and this quantity “measures” information. Information is a basic aspect of physical reality, and thus the entropy function will pop up no matter where one looks.
This is a simple function with an extraordinary number of nice properties, linking probability, convexity, dynamical systems, and many other subjects. In otherwords, it is something like the polylogarithm; and for the same reasons like the polylogarithm it pops up all over the place.

The first approach is, I think, how a typical physicist thinks about it; the second is reminds me of of John von Neumann’s recommendation that Shannon call $S$ the “entropy”, because

no one really knows what entropy is, so in a debate you will always have the advantage.

No one knows what the polylogarithm is, either. Moreover, they are closely related, as Kontsevich’s appendix On Poly(ana)logs I explains; the finite-sum truncations of the polylogarithm function are mod- $p$ analogs of the entropy! This is further discussed in this entertatining nCafe post. Note that the content of the “Poly(ana)log” paper can is essentially the derivation of functional equations analogous to the functional equations that entropy functionals satisfy; and Shannon’s derivation of $S$ was by the functional equation that any ``entropy-like’’ quantity should satisfy. Yet, without all the high-concept ideas going into the “Poly(ana)log paper”, even Kontsevich didn’t know how to derive these functional equations; and Kontsevich probably found his identities because he was thinking hard about the mathematics of quantum field theory.

What’s the physical point of entropy mod- $p$ ? Well, none, for now. But from this example I’ll add one more conclusion to my list:

D) Part of the point of mathematics is to develop conceptual tools which let humans navigate the horrifying maze created by the interrelationships between the simplest mathematical constructions possible. So, to prove more identities between things related to polylogarithms, you think about algebraic-K theory. By shuttling insight between extremely simple structures, which can be interpreted in many different ways but seem unstructured, and extremely abstract theory, which is very rigid but can be hard to gain intuition for, one learns more about the entire forest of mathematics.

From this perpsective, it seems inevtiable that when non-mathematicians try to use formal constructions in their work, they will fall into this web of relationships and be able to get farther much faster by going down paths that have alrady been set by mathematicians past. So one gets “mysterious relationships” between abstract mathematics and other subjects; but really, this is just that one cannot really navigate simple mathematics without all the mathematical abstraction built around it, and simple mathematics is inevitable.