Pure Maths Colloquium: Combinatorics and toric topology of Stasheff polytopes
11 February 2008 16:15 in CM221
The Stasheff polytopes $K_n, n>2,$ appeared in the Stasheff
paper ``Homotopy associativity of H-spaces'' (1963) as the spaces of
homotopy parameters for maps determining associativity conditions
for the products $a_1 ...a_n, n>2$.
Stasheff polytopes are the focus of attention of various research
areas. Nowadays they have become well-known due to applications of
operad theory in physics. There is a growing number of different
approaches, such that bracketing, polygon dissection, plane trees,
intervals and so on, which result in Stasheff polytopes.
We will describe combinatorics and toric topology of Stasheff
polytopes using several constructions of these polytopes.
We will show that the two-parameter generating function $U(t,x)$,
enumerating the number of $k$-dimensional faces of the $n$-th
Stasheff polytope, satisfies the famous Burgers-Hopf equation
We will discuss some applications of this result including an
interpretation of the Dehn--Sommerville relations in terms of the
Cauchy problem, and the Cayley formula in terms of conservation
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