On Quasiminimal Excellent Classes

A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_w1,w(Q)-definability assumption may be dropped, and each class is determined by its model of dimension aleph_0.Comment: 16 pages. v3: correction to the statement of corollary 5.

Axioms for infinite matroids

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig

A Topological Representation Theorem for Oriented Matroids

We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{\"o}nflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.Comment: 21 pages, 4 figure

Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

Poisson spaces with a transition probability

The common structure of the space of pure states $P$ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function $p:P\times P-> [0,1]$, with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(\rho,\sigma)=\dl_{\rho\sigma}, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of $p$, and by the property that the irreducible components of $P$ as a transition probability space coincide with the symplectic leaves of $P$ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz ({\em Acta Math.} {\bf 144} (1980) 267-305) and F.W. Shultz ({\em Commun.\ Math.\ Phys.} {\bf 82} (1982) 497-509), we give axioms guaranteeing that $P$ is the space of pure states of a unital $C^*$-algebra. We give an explicit construction of this algebra from $P$.Comment: 23 pages, LaTeX, many details adde