Algorithms in algebra and topology

Información del proyecto

GroupsComputability

Identificador del acuerdo de subvención: 659102

DOI

10.3030/659102

Proyecto cerrado

Fecha de la firma de la CE 6 Marzo 2015

Fecha de inicio 1 Octubre 2015

Fecha de finalización 30 Septiembre 2017

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 195 454,80

Aportación de la UE

€ 195 454,80

Coordinado por

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE
United Kingdom

Objetivo

Group theory is the study of symmetry in mathematical objects, such as rotations of geometric shapes.
Groups help us understand the underlying structure of mathematical objects by revealing their symmetries.
To understand groups we need an efficient way to describe them. Some groups admit a finite presentation;
a finite set of building blocks, along with a finite collection of rules on when we can substitute one set
of blocks for another. These descriptions are convenient. However, results in algebra and logic show
that such descriptions are not always suitable to work with, as certain problems (e.g. the word problem,
of deciding if two distinct collections of blocks represent the same group element) are incomputable; no
computer can be built to always answer this. We can embed incomputable problems from groups into
geometry, to show that the homeomorphism problem, of recognising if two geometric shapes are equivalent
under smooth deformation, is incomputable in all dimensions above three. Thus we can't computationally
classify geometric shapes in higher dimensions; we can't identify the unique distinguishing features of
each shape. The study of generic computability (problems which can be computed most of the time) is
a useful area in mathematics. Conversely, showing a problem can't be computed most of the time gives
rise to applications in cryptography: generically incomputable problems are an excellent tool in the theory
behind cryptosystems. This proposal will deal with incomputable and generically incomputable problems.
We will investigate certain problems in group theory to determine if they are computable, or generically
computable, or neither. We will apply these results to particular classess of higher-dimensional geometric
objects, identifying whether certain problems relating to them are computable or not. The project will be
carried out at the University of Cambridge, under the supervision of Dr. Henry Wilton.

Ámbito científico (EuroSciVoc)

• ciencias naturales › informática y ciencias de la información › internet
• ciencias naturales › matemáticas › matemáticas puras › topología
• ciencias naturales › informática y ciencias de la información › seguridad informática › criptografía
• ciencias naturales › matemáticas › matemáticas puras › álgebra
• ciencias naturales › matemáticas › matemáticas puras › geometría

Programa(s)

• H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions PROGRAMA PRINCIPAL
  
• H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility
  

Tema(s)

• MSCA-IF-2014-EF - Marie Skłodowska-Curie Individual Fellowships (IF-EF)
  

Régimen de financiación

MSCA-IF-EF-ST - Standard EF

Convocatoria de propuestas

H2020-MSCA-IF-2014

Coordinador

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE

• Aportación neta de la UEn: € 195 454,80
• Dirección: TRINITY LANE THE OLD SCHOOLS ; CB2 1TN CAMBRIDGE ; Reino Unido
• Región: East of England › East Anglia › Cambridgeshire CC
• Tipo de actividad: Higher or Secondary Education Establishments
• Coste total: € 195 454,80