Are there any Applications of Abstract Algebra in Engineering?

Question

As the title suggests, I was wondering whether there are any applications of abstract algebra in the engineering disciplines - and if so, what these are (not including basic linear algebra here, as that one seems trivial to answer - this is about Abstract Algebra in the sense of Dummit & Foote).

And, for those who might be knowledgeable on this particular topic, are there any areas (of engineering, that is) where there's potential for abstract algebra to make a difference, or where ongoing research is already pointing to promising results?

Materials science, where the symmetry groups of crystal structures are critical. — John Hughes, May 05 '18 at 18:05
All of the above are the sort of suggestions and answers I'm hoping for. — Ius Klesar, May 05 '18 at 18:07
Lie groups show up in mechanics, Lie brackets show up in control theory. — ITA, May 05 '18 at 18:12
Also yes, research is ongoing in using abstract algebra in the are of dynamical systems, for example: https://arxiv.org/abs/1602.01017 — ITA, May 05 '18 at 18:14
@John Hughes Good answer! I think one of the most fundamental errors in the teaching of advanced mathematics is the omission of the crystallographic groups from intro abstract algebra courses. — C Monsour, May 05 '18 at 18:39

lhf · Answer 1 · 2018-05-05T18:57:23.970

1

Algebraic Geometry is useful in Computer-Aided Geometric Design. See for instance this paper:

Algebraic Geometry for Computer-Aided Geometric Design, by Thomas W. Sederberg and Ronald N. Goldman, in IEEE Computer Graphics and Applications, 1986.

The whole field of Computer-Aided Geometric Design is very mathematical.

See also this book:

Solving Systems of Polynomial Equations, by Bernd Sturmfels, AMS, 2002

edited May 05 '18 at 18:57

answered May 05 '18 at 18:39

lhf

221,500

1

AG has also been applied to filter design, e.g. Solving the Selesnick-Burrus Filter Design Equations Using Computational Algebra and Algebraic Geometry – May 05 '18 at 18:42
4

As someone who has been doing computer graphics for a very long time, I'd have to say that the applications of abstract algebra in CAGD are pretty limited (although to the degree that you think of the Fourier Transform as part of group theory, there is a nice application of it to prove (limited) smoothness of subdivision surfaces). I was lead author of the most recent edition of one of the standard computer graphics texts, and despite my strong mathematical leanings, I never had occasion to mention abstract algebra. – John Hughes May 05 '18 at 18:48

Are there any Applications of Abstract Algebra in Engineering?

1 Answers1