The Poincare polynomial for the cohomology of the configuration space [F.sub.n]([R.sup.d]) is equal to
It follows that the Poincare polynomial (where [beta]j are Betti numbers) of [S.sup.c.sub.n]([R.sup.2])/SO(2) is
where Poin(M(A[(I).sub.C]), t) is the Poincare polynomial of the complement M(A[(I).sub.C]) of A[(I).sub.C] and the nonnegative integers [d.sub.1], ..., [d.sub.l] coincide with DP (I).
Then the Poincare polynomial of the topological space M (A) splits as
These proceedings of the October 2005 conference held in Hanoi include survey and research articles reflecting current interest in a range of topics, including a survey on Zariski pairs and another on a categorical construction of Lie algebras, and research on elliptical parameters and defining equations for elliptic fibrations on a Kummer surface, characterization of the rational homogeneous space association with a long simple root by its variety of minimal rational tangents, maximal divisorial sets in arc spaces, two non-conjugate embeddings into Cremora Group II, the Castelnuovo-Severi inequality for a double covering, a Poincare polynomial
of a class of signed complete graphic arrangements, singularities of dual varieties in Characteristic 2, and Castelnuovo-Well lattices.
7] the following result linking a q-analogue [A.sub.w](q) (whose definition we omit) of A[O.sub.w] to the Poincare polynomial
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (here the sum is over the permutations u in the Bruhat interval [id, [w.sub.[lambda]]]).
But the idea is interesting, and can (sometimes) be applied to a finer topological invariant: the sequence of Betti numbers, or more precisely, the Poincare polynomial
In particular, if W is the Weyl group of R, and l is the length function on W, then the Poincare polynomial
P(q) = [[summation].sub.w[member of]W][q.sup.l(w)] = [[PI].sup.l.sub.i=1][[[m.sub.i] + 1].sub.q], where [[m].sub.q] is the q-integer (1 + q + ...
The cohomology ring of the complement V (A) has Poincare polynomial
It is well known that, if W is any finite Coxeter or affine Weyl group, [F.sub.[upsilon]](q) is the intersection homology Poincare polynomial
of the Schubert variety indexed by [upsilon] (see Kazhdan and Lusztig (1980)).
The Poincare polynomial
[P.sub.w](q) is the rank generating function for the order ideal of w.
Therefore, the homology Poincare polynomial
for [[pi].sup.-1] ([e.sub.u]) is