Polylogarithms in arithmetic and geometry

The classical polylogarithms were invented in correspondence of Leibniz with Joh. Bernoulli in 1696. They are defined inductively: $$Li_n(z)\coloneq \int^z_0Li_{n-1}(t)\frac{dt}t,\ Li_1(z)=-\log(1-z)$$ The dilogarithm was studied widely during the XIXth century (Spence, Abel, Kummer...) and appeared again in the id of 70th in works of Gabrielov-Gelfand-Losik, Bloch and Wigner.

We will discuss how and why the classical $n$-logarithms appear in such different contexts as:

1. Explicit formulas for special values of Dedekind $\zeta$-functions $\zeta_F(s)$ of an arbitrary number fields $F$ at $s=n$ (Zagier's conjecture; proved for $n=2,3$).
2. Explicit formulas special values of Hasse-Weil $L$-functions $L(E,s)$ of an elliptic curve over number fields at $s=n$ (Bloch: $n=2$ case; Deninger's conjecture for $n=3$, proved for elliptic modular curves over ${\bf Q}$ for $n\neq 4$).
3. More generally, explicit formulas for Be\u\i linson's regulator $r_{Be}\colon K_{2n-2}(X)\to H^2_{\scr D}(X/R,R(n))$ for curves over number fields generalizing symbol modére of Be\u\i linson and Deligne for $n=2$ (proved for $n=3,4)$).
4. Description of Quillen's $K$-groups $K_n(F)\otimes {\bf Q}$ of an arbitrary field $F$ by generators and relations in the spirit of Milnor's definition of algebraic $K$-theory. (Known for $n=2,3$, considerable evidence for $n\leq 6$).
5. Closely related problem of explicit construction of motivic complexes conjectured by Be\u\i linson and Lichtenbaum.
6. The structure of the motivic Galois group of the (hypothetical) category of mixed Tate motives.
7. Explicit formulas for measurable cocycles of the Lie group ${\rm GL}_n({\bf C})$.
8. Computation of volumes of odd-dimensional hyperbolic manifolds.