Toric Topology of Stasheff Polytopes

The Stasheff polytopes $K_n$, $n\ge 3$, first arose in his paper ``Homotopy associativity of $H$-spaces'' (1963) as the space of homotopy parameters for maps determining associativity conditions for a product $a_1 \cdots a_n,\; n \geqslant 3$; they were defined via binary bracketings of the formal monomials $a_1 \cdots a_n$. Stasheff polytopes are in the limelight of several research areas, recently especially in connection with physical applications of operad theory. We will describe geometry and combinatorics of Stasheff polytopes using the several different ways to introduce these polytopes and the methods and results of toric topology. We will show that the two-parameter generating function $U(t,x)$ that enumerates the number of $k$-dimensional faces of the $n$-th Stasheff polytope satisfies the famous Burgers--Hopf equation $U_t=UU_x$. We will discuss the applications of this result including an interpretation of Dehn--Sommerville relations in the term of Cauchy problem, and Cayley formula in the term of conservation laws