Chern Classes of the Moduli Stack of Curves

Here we calculate the Chern classes of ${\bar {\mathcal M}}_{g,n}$, the moduli stack of stable n-pointed curves. In particular, we prove that such classes lie in the tautological ring.Comment: submitted preprin

Mutant knots with symmetry

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m<6 in general, and for all choices of m when the satellite is based on a cable knot pattern. We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.Comment: 15 page

On second order Thom-Boardman singularities

In this paper we derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities \Sigma^{i,j}. The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.Comment: 15 pages, 1 figure; minor updates and correction

The stability of the Kronecker products of Schur functions

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.Comment: 16 page

Resolving mixed Hodge modules on configuration spaces

Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a formula for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in the Grothendieck group of the category of mixed Hodge structures) of the moduli space M_{1,n}. In a sequel to this paper, this is applied in the calculation of the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}.Comment: 25 pages. amslatex-1.2, pb-diagram and lamsarrow. There are a number of corrections from the first versio