It was conjectured in [5] and proved by Mark Haiman in [13] that the Frobenius Characteristic of the S n Module of Diagonal Harmonics is none other than ∇e n . Here ∇ is the symmetric function operator introduced in [1] with eigen-functions the modified Macdonald basis {H μ } μ . The Shuffle Conjecture [12] expresses the scalar product ∇e n , h μ1 h μ2 • • • h μ k as a weighted sum of Parking Functions on the n × n lattice square which are shuffles of k increasing words. In [10] Jim Haglund succeeded in proving the k = 2 case of this conjecture. In this paper we give a new and more direct proof of the combinatorial part of Haglund's argument and obtain a substantial reduction in the complexity of the symmetric function part. * Work carried out under NSF support.

Abstract. We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to which a variety is not smooth that point. We conjecture a closed formula in the case of curves with one singular point, and we provide explicit formulas for this series in the case of the cusp and the node. Using the work of Denef and Loeser, one can show that this series will often be rational.These ideas were obtained through extensive calculations in Sage. Thus, we include a Sage script which was used in these calculations. It computes the affine arc spaces ∇nX provided that X is affine, n is a fat point, and the ground field is of characteristic zero. Finally, we show that the auto Poincaré series will often be rational as well and connect this to questions concerning new types of motivic integrals.

We develop further the theory of integrable functions within the theory of relative simplicial motivic measures. We provide a primitive change of variables formula for this theory.

We develop a universal theory of relative simplicial motivic measures.