Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

Integrals and Valuations

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.Comment: Submitted for publication 15/05/0

A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields

In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be extended to face these {\it kinematical} non-linearities squarely. We first present a pedagogical account of this problem and then suggest an avenue for its resolution.Comment: 27 pages, CGPG-94/8-2, latex, contribution to the Cambridge meeting proceeding

Constructive Matrix Theory for Higher Order Interaction

This paper provides an extension of the constructive loop vertex expansion to stable matrix models with interactions of arbitrarily high order. We introduce a new representation for such models, then perform a forest expansion on this representation. It allows to prove that the perturbation series of the free energy for such models is analytic in a domain uniform in the size N of the matrix. Our method applies to complex (rectangular) matrices. The extension to Hermitian square matrices, which was claimed wrongly in the first arXiv version of this paper, is postponed to a future study.Comment: 44 pages, 9 figure

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

High order symplectic integrators for perturbed Hamiltonian systems

We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O(\tau^p\epsilon +\tau^2\epsilon^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O(\tau^p\epsilon +\tau^4\epsilon^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.Comment: 24 pages, 6 figurre