Related papers: Computable structures on topological manifolds

Topological Data Analysis (TDA) combines computational topology and data science to extract and analyze intrinsic topological and geometric structures in data set in a metric space. While the persistent homology (PH), a widely used tool in…

In this paper, we consider a simple class of stratified spaces -- 2-complexes. We present an algorithm that learns the abstract structure of an embedded 2-complex from a point cloud sampled from it. We use tools and inspiration from…

By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…

Manifolds and fiber bundles, while superficially different, have strong parallels; in particular, they are both defined in terms of equivalence classes of atlases or in terms of maximal atlases, with the atlases treated as mere adjuncts.…

For positive integers $m$ and $s$, let $\mathbf{m}_s$ stand for the $s$-th tuple $(m,\ldots,m)$. We show that, for large enough $s$, the higher topological complexity $TC_s$ of an even dimensional real projective space $RP^m$ is…

Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the {\em computationally dense\/} ones) are seen to be the ones…

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated…

A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…

Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] the authors define a topology on $X$ by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction…

(Talk presented at the XVth Workshop on Geometric Methods in Physics, Quantizations, Deformations and Coherent States, in Bialowieza, Poland, July 1-7, 1996.) The aim of this article is to introduce some basic notions of Topological Quantum…

We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if $p \geq 1$ is a computable real, and if $\Omega$ is a nonzero, non-atomic, and separable measure space, then every…

Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…

This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…

The complexity of large-scale distributed systems, particularly when deployed in physical space, calls for new mechanisms to address composability and reusability of collective adaptive behaviour. Computational fields have been proposed as…

We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…

A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are…

We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in…

We present a generalization of standard Turing machines based on allowing unusual tapes. We present a set of reasonable constraints on tape geometry and classify all tapes conforming to these constraints. Surprisingly, this generalization…

This is a brief introduction to the basic concepts of topology. It includes the basic constructions, discusses separation properties, metric and pseudometric spaces, and gives some applications arising from the use of topology in computing.

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…