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.
* Induction formula for singularities in families. The central result of the present work is the induction formula of . A similar formula has been proved by Kazarian, Lando and Zvonkine for classes of singularities in families of genus 0 stable maps (see @cite_28 ). Their formula contains only the genus 0 part of our induction formula.
@cite_28
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We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincar 'e dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the above properties. Curves of genus g which admit a map to CP1 with specified ramification profile mu over 0 and nu over infinity define a double ramification cycle DR_g(mu,nu) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR_g(mu,nu) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR_g(mu,nu) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain's formula in the compact type case. When mu and nu are both empty, the formula for double ramification cycles expresses the top Chern class lambda_g of the Hodge bundle of the moduli space of stable genus g curves as a push-forward of tautological classes supported on the divisor of nonseparating nodes. Applications to Hodge integral calculations are given. Relations among tautological classes on the moduli space of stable curves are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r=4 relations are used to bound the Betti numbers of the tautological ring of the moduli of nonsingular curves. At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class. In the Appendix (with F. Janda), a conjecture relating the r=0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented. In this paper we consider particular equational formulae where equality = AC is the congruence induced by a set of associative-commutative axioms. The formulae we are interested in have the form t ? ACt1 ??t ? ACtn and are usually known as complement problems. To solve a complement problem is to find an instance of t which is not an instance of any ti modulo associativity-commutativity. We give a decision procedure based on tree automata which solves these formulae when all the ti1s are linear. We show that this solution also gives a decision of inductive reducibility modulo associativity and commutativity in the linear case and we give several extensions of this approach. Then, we define a new class of tree automata, called conditional tree automata which recognize sets of generalized terms, i.e terms written as multisets, and where the application of a rule depends on the satisfiability of a formula of Presburger's arithmetic. This class of tree automata allows one to solve non-linear complement problems when all occurrences of a non-linear variable occur under the same node (in flattened terms). This solution also provides a procedure to decide inductive reducibility modulo associativity-commutativity in the same case.