Relations in the cohomology ring of the moduli space of flat (2 + 1)-connections on a Riemann surface.

We consider the moduli space of flat (2 + 1)-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalizing a conjecture of Newstead.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Convexity of the moment map image for torus actions on -symplectic manifolds.

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a -symplectic manifold.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Signature quantization.

We associate to the action of a compact Lie group G on a line bundle over a compact oriented even-dimensional manifold a virtual representation of G using a twisted version of the signature operator. We obtain analogues of various theorems in the more standard theory of geometric quantization.

The Euler-Maclaurin formula for simple integral polytopes.

We give a Euler-Maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope. Our proof uses elementary methods.