The Computability Problem in Algebraic Topology

The Computability Problem in Algebraic Topology

At first, and this has been immediately noted, this result ensures the durability of the mathematical job, which is not inconsiderable. No finite “system", in the broadest sense of the term, can potentially cover the whole mathematical field. So it can be considered that Church and Turing have proved that mathematicians will always be able to discover new playing fields and their creativitywill...

The Arf-kervaire Invariant Problem in Algebraic Topology: Introduction

This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper in this volume, The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof. A rigorous account can be found in our preprint Th...

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

Topology and Computability Thesis Proposal

In this thesis I will relate standard computability to computability of reals|and related spaces|and link these notions with denotational semantics of computer languages. The framework for the proposed study was worked out in collaboration with Steve Awodey, Lars Birkedal and Dana Scott. The rst section gives a motivation for our setup and indicates how standard computability ts in it. The subs...

Randomness, Computability, and Algebraic Specifications