Mathematics

We determine the Chow ring (with Q-coefficients) of M 6 by showing that all Chow classes are tautological. (In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective.) We stratify the moduli space into locally closed strata which are group quotients, and use a theorem of Vistoli to show that their Chow rings are generated by Chern classes of natural vector bundles. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus six case relies in addition on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank five vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank six vector bundle of quadrics cutting out the canonical curve.

We determine the Chow ring (with Q-coefficients) of M 6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank 6 vector bundle of quadrics cutting out the canonical curve.

A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems) met in practice. A f.i.p. theory FS 0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS 0 is a conservative extension of PRA. The aims of this work are (i) conceptual, (ii) pedagogical and (iii) practical. The system FS 0 serves under (i) and (ii) as a theoretical framework for the formalization of metamathematics. The general approach may be used under (iii) for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way.

Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. 1 It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.

Questions of definedness are ubiquitous in mathematics. Informally, these involve reasoning about expressions which may or may not have a value. This paper surveys work on logics in which such reasoning can be carried out directly, especially in computational contexts. It begins with a general logic of "partial terms", continues with partial combinatory and lambda calculi, and concludes with an expressively rich theory of partial functions and polymorphic types, where termination of functional programs can be established in a natural way.

In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type.