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GroupsComputability

Algorithms in algebra and topology

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EC-Contrib. €

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Project "GroupsComputability" data sheet

The following table provides information about the project.

Coordinator
THE CHANCELLOR MASTERS AND SCHOLARSOF THE UNIVERSITY OF CAMBRIDGE 

Organization address
address: TRINITY LANE THE OLD SCHOOLS
city: CAMBRIDGE
postcode: CB2 1TN
website: www.cam.ac.uk

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Project website https://www.dpmms.cam.ac.uk/
 Total cost 195˙454 €
 EC max contribution 195˙454 € (100%)
 Programme 1. H2020-EU.1.3.2. (Nurturing excellence by means of cross-border and cross-sector mobility)
 Code Call H2020-MSCA-IF-2014
 Funding Scheme MSCA-IF-EF-ST
 Starting year 2015
 Duration (year-month-day) from 2015-10-01   to  2017-09-30

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    THE CHANCELLOR MASTERS AND SCHOLARSOF THE UNIVERSITY OF CAMBRIDGE UK (CAMBRIDGE) coordinator 195˙454.00

Map

 Project objective

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.

 Publications

year authors and title journal last update
List of publications.
2016 Maurice Chiodo, Rishi Vyas
Torsion length and finitely presented groups
published pages: , ISSN: , DOI:
2019-06-17
2016 Maurice Chiodo, Michael Hill
Preserving torsion orders when embedding into groups with small finite presentations
published pages: , ISSN: , DOI:
2019-06-17
2016 Maurice Chiodo, Zachiri McKenzie
X-torsion and universal groups
published pages: , ISSN: , DOI:
2019-06-17

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