Michèle Vergne

Residue formulae for vector partitions and Euler-MacLaurin sums

Let V be an n-dimensional real vector space endowed with a rank-n lattice Γ . The dual lattice Γ ∗ = Hom(Γ.Z) is naturally a subset of the dual vector space V ∗. Let Φ = [β1, β2, . . . , βN ] be a sequence of not necessarily distinct elements of Γ ∗, which span V ∗ and lie entirely in an open halfspace of V ∗. In what follows, the order of elements in the sequence will not matter. The closed cone C(Φ) generated by the elements of Φ is an acute convex cone, divided into open conic chambers by the (n− 1)-dimensional cones generated by linearly independent (n−1)-tuples of elements of Φ . Denote by ZΦ the sublattice of Γ ∗ generated by Φ . Pick a vector a ∈ V ∗ in the cone C(Φ), and denote by ΠΦ(a) ⊂ R+ the convex polytope consisting of all solutions x = (x1, x2, . . . , xN) of the equation ∑Nk=1 xkβk = a in nonnegative real numbers xk . This is a closed convex polytope called the partition polytope associated to Φ and a. Conversely, any closed convex polytope can be realized as a partition polytope. If λ ∈ Γ ∗, then the vertices of the partition polytope ΠΦ(λ) have rational coordinates. We denote by ιΦ(λ) the number of points with integral coordinates in ΠΦ(λ). Thus ιΦ(λ) is the number of solutions of the equation ∑N k=1 xkβk = λ in nonnegative integers xk . The function λ → ιΦ(λ) is called the vector partition function associated to Φ . Obviously, ιΦ(λ) vanishes if λ does not belong to C(Φ) ∩ZΦ .

Arrangement of hyperplanes. I : Rational functions and Jeffrey-Kirwan residue

Consider the space RΔ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of RΔ as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and of residue to the space RΔ, and we describe their relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

Toric reduction and a conjecture of Batyrev and Materov

We present a new integration formula for intersection numbers on toric quotients, extending the results of Witten, Jeffrey and Kirwan on localization. Our work was motivated by the Toric Residue Mirror Conjecture of Batyrev and Materov; as an application of our integration formula, we obtain a proof of this conjecture in a generalized setting.

How to Integrate a Polynomial over a Simplex

This paper starts by settling the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We explore our algorithms with some experiments. We conclude the article with extensions to other polytopes and discussion of other available methods. 1.

Counting Integer Flows in Networks

AbstractnThisn paper discusses analytic algorithms and software for thenenumeration of all integer flows inside a network. Concretenapplications abound in graph theory, representation theory, andnstatistics. Our methods are based on the study of rational functionsnwith poles on arrangements of hyperplanes; they surpass traditionalnexhaustive enumeration and can even yield formulas when the input datancontains some parameters. We also discuss the calculation of chambersnin detail because it is a necessary subroutine.

Volume computation for polytopes and partition functions for classical root systems

This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.

Box splines and the equivariant index theorem

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distributions, on the dual of the group G or on the dual of its Lie algebra. The morphism from K-theory to cohomology is analyzed and the multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semidiscrete convolution with a box spline. Finally, the multiplicities of the index of a G-transversally elliptic operator on M are determined using the infinitesimal index of the symbol.

A proof of Bismut local index theorem for a family of Dirac operators

On donne une demonstration du theoreme de Bismut basee sur le developpement classique des noyaux de la chaleur. On exprime le noyau de la chaleur du laplacien dun fibre vectoriel comme une moyenne sur le groupe dholonomie

Le centre de l'algébre enveloppante et la formule de Campbell-Hausdorff☆

Resume Si g est une algebre de Lie, nous definissons des fonctions F ( x , y ) et G ( x , y ) sur g⊕g a valeurs dans g telles que x + y − log (e x e y ) = (e ad x ) −1 F ( x , y +(1−e −ad y G ( x , y ). Si g est une algebre de Lie quadratique, nous prouvons une identite pour la trace de la matrice (ad x )∂ x F + (ad y )∂ y G . Cette identite est conjecturee dans [4] pour toute algebre de Lie, et demontree si g est resoluble. Elle implique (voir [4]) le prolongement naturel de lisomorphisme de Duflo [2] aux algebres de convolution de distributions invariantes sur le groupe G et sur lalgebre de Lie g.

Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A.xa0Barvinok in the unweighted case (i.e., h≡1). In contrast to Barvinok’s method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach, we report on computational experiments which show that even our simple implementation can compete with state-of-the-art software.