What is the correct combinatorial object to encode a linear representation?  Many shadows of this problem have been studied:moment polytopes, Duistermaat-Heckman measures, Okounkov bodies.  We suggest that already in very simple cases these miss a crucial feature.  The ring theory, as opposed to just the linear algebra, of the group action on the coordinate ring, depends on some non-trivial lattice geometry and an associated filtration.  Some striking similarities to, and key differences from, the theory of toric varieties ensue.  Finite and non-finite generation phenomena emerge naturally.  We discuss motivations from, and applications to, questions in the effective geometry of moduli of curves.

 The Sen conjecture, made in 1994, makes precise predictions about the existence of L^2 harmonic forms on the monopole moduli spaces. For each positive integer k, the moduli space M_k of monopoles of charge k is a non-compact smooth manifold of dimension 4k, carrying a natural hyperkaehler metric.  Thus studying Sen’s conjectures requires a good understanding of the asymptotic structure of M_k and its metric.  This is a challenging analytical problem, because of the non-compactness of M_k and because its asymptotic structure is at least as complicated as the partitions of k.  For k=2, the metric was written down explicitly by Atiyah and Hitchin, and partial results are known in other cases.  In this talk, I shall introduce the main characters in this story and describe recent work aimed at proving Sen’s conjecture.

Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate.