The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT).  Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. Methods from non-reductive GIT allow us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves of fixed singularity type.

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.
 

In this talk we will motivate and discuss several problems and results in harmonic analysis that involve some arithmetic or discrete structure. We will focus on pioneering work of Bourgain on discrete restriction theorems and pointwise ergodic theorems for arithmetic sets, their modern developments and future directions for the field.

Joint work with Wolfgang Hackbusch and Daniel Kressner

(joint work with Sergei Starchenko)

Let p:C^n ->A be the covering map of a complex abelian variety and let X be an algebraic variety of C^n, or more generally a definable set in an o-minimal expansion of the real field. Ullmo and Yafaev investigated the topological closure of p(X) in A in the above two  settings and conjectured that the frontier of p(X) can be described, when X is algebraic as finitely many cosets of real sub tori of A, They proved the conjecture when dim X=1. They make a similar conjecture for X definable in an o-minimal structure.

In recent work we show that the above conjecture fails as stated, and prove a modified version,  describing the frontier of p(X) as finitely many families of cosets of subtori. We prove a similar result when X is a definable set in an o-minimal structure and p:R^n-> T is the covering map of a real torus.  The proofs use model theory of o-minimal structures as well as algebraically closed valued fields.

In various research fields such as machine learning, compressed sensing and operations research, optimization problems which seek sparsity of solutions by the cardinality constraint or rank constraint are studied. We formulate such problems as DC (Difference of two Convex functions) optimization problems and apply DC Algorithm (DCA) to them. While a subproblem needs to be solved in each DCA iteration, its closed-form solution can be easily obtained by soft-thresholding operation. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods.
This is a joint work with J. Gotoh (Chuo Univ.) and K. Tono (U. Tokyo).