We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
We consider the two following functions: Both @math and @math are polynomials for large values of @math (this result is due to Aaron Pixton, see @cite_32 and @cite_1 ). We denote by @math and @math the asymptotic polynomials. The two following conjectures have been proposed.
@cite_1 @cite_32
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We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type. We study compositions of the integer @math in which the first part, successive differences, and the last part are constrained to lie in prescribed sets @math , respectively. A simple condition on @math guarantees that the generating function @math has only a simple pole on its circle of convergence and this at @math , a function independent of @math and @math . Thus the number of compositions is asymptotic to @math for a suitable constant @math . We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of @math , such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If @math has only finitely many positive elements @math , or finitely many negative elements @math , then the largest part and number of distinct part sizes are almost surely @math . On the other hand, when both @math and @math have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely @math , and we give sufficient conditions for the largest part to be almost surely asymptotic to @math .