An Intuitionistic Formula Hierarchy Based on High-School Identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism

Spectral Curves, Opers and Integrable Systems

We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. The spectral data are principal G-bundles on an algebraic curve, equipped with an abelian reduction near one point. The flows on the spectral side come from the action of a Heisenberg subgroup of the loop group. The differential data are flat connections known as opers. The flows on the differential side come from a generalized Drinfeld-Sokolov hierarchy. Our isomorphism between the two sides provides a geometric description of the entire phase space of the Drinfeld-Sokolov hierarchy. It extends the Krichever construction of special algebro-geometric solutions of the n-th KdV hierarchy, corresponding to G=SL(n). An interesting feature is the appearance of formal spectral curves, replacing the projective spectral curves of the classical approach. The geometry of these (usually singular) curves reflects the fine structure of loop groups, in particular the detailed classification of their Cartan subgroups. To each such curve corresponds a homogeneous space of the loop group and a soliton system. Moreover the flows of the system have interpretations in terms of Jacobians of formal curves.Comment: 64 pages, Latex, final version to appear in Publications IHE

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

Higher-dimensional Algebra and Topological Quantum Field Theory

The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n-dimensional unitary extended TQFTs as weak n-functors from the "free stable weak n-category with duals on one object" to the n-category of "n-Hilbert spaces". We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction.Comment: 36 pages, LaTeX; this version includes all 36 figure