Corrado De Concini

Nested sets and Jeffrey-Kirwan residues

For the complement of a hyperplane arrangement we construct a dual homology basis to the no-broken-circuit basis of cohomology. This is based on the theory of wonderful embeddings and nested sets developed in [4]. Our result allows us to express the so-called Jeffrey-Kirwan residues in terms of integration on some explicit geometric cycles.

Normality and non Normality of certain semigroups and orbit closures

Given a representation ρ: G → GL(N) of a semisimple group G, we discuss the normality or non normality of the cone over ρ(G) using the wonderful compactification of the adjoint quotient of G and its projective normality [K]. These methods are then used to discuss the normality or non normality of certain other orbit closures including determinantal varieties.

A curious identity and the volume of the root spherical simplex

We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots.

The Partition Functions

The main purpose of this chapter is to discuss the theory of Dahmen–Micchelli describing the difference equations that are satisfied by the quasipolynomials that describe the partition function (mathcal{T}_X) on the big cells. These equations allow also us to develop possible recursive algorithms.

Approximation by Splines

In this chapter we want to give a taste to the reader of the wide area of approximation theory. This is a very large subject, ranging from analytical to even engineering-oriented topics. We merely point out a few facts more closely related to our main treatment. We refer to [70] for a review of these topics.

Approximation Theory I

In this chapter we discuss an approximation scheme as in [33] and [51], that gives some insight into the interest in box splines, which we will discuss presently.

Modules over the Weyl Algebra

All the modules over Weyl algebras that will appear are built out of some basic irreducible modules, in the sense that they have finite composition series in which only these modules appear. It is thus useful to give a quick description of these modules. Denote by F the base field (of characteristic 0) over which V,U := V* are finite-dimensional vector spaces of dimension s. We can take either (F = mathbb{R}) or (F = mathbb{C}).

Fourier and Laplace Transforms

This short chapter collects a few basic facts of analysis needed for the topics discussed in this book.

Differential and Difference Equations

The purpose of this chapter is to recall standard facts about certain special systems of differential equations that admit, as solutions, a finite-dimensional space of exponential polynomials. The theory is also extended to difference equations and quasipolynomials.

R X as a D-Module

In this chapter, the word D–module is used to denote a module over one of the two Weyl algebras W(V),W(U) of differential operators with polynomial coefficients on V,U respectively. The purpose of this chapter is to determine an expansion in partial fractions of the regular functions on the complement of a hyperplane arrangement. This is essentially the theory of Brion–Vergne (cf. [28], [7]). We do it using the D–module structure of the algebra of regular functions. Finally, by inverse Laplace transform all this is interpreted as a calculus on the corresponding distributions.