Talking about topology and engineering, I remember
Raoul Bott studied engineering when he was a student.
His thesis at Carnegie Mellon University is entitled
“Electrical Network Theory”.

And now, Robert Ghrist. He is very actively trying to
apply mathematics, including algebraic topology, to
problems in engineering. A lot of papers and
expositions can be downloaeded from his website. For
example, he wrote an article
[
dSG07b
]
on his work on sensor networks.

I had a chance to listen to Ghrist’s talk
during the conference on
braids
in Singapore in 2007. His talk and presentation
techniques are impressive. Furthermore, he said that
he is advertising algebraic topology to peaple in
engineering, since algebraic topology is useful.

Applications of algebraic topology discovered by
Ghrist and his collaborators include the following.

In their study of sensor networks, a classical way of
constructing a
simplicial complex
from a covering, called the Rips complex, used by
Vietoris in 1920’s is used.

Daniel Cohen and Pruidze
[
CP
]
refer to
[
Lat91
,
Sha97
]
for motion planning.

Adler and his collaborators
[
ABB
^{
+
}
]
used
persistent homology
.

It is interesting that they used the phrase
“applied algebraic topology” in their
paper. It seems the topics in this page are forming a
new research area.

[ABB
^{
+
}
]
Robert J. Adler, Omer Bobrowski,
Matthew S. Borman, Eliran Subag, and Shmuel
Weinberger. Persistent Homology for Random Fields
and Complexes,
arXiv:1003.1001
.

[CP]
Daniel C. Cohen and Goderdzi Pruidze. Motion
planning in tori,
arXiv:math.GT/0703069
.

[dSG07a]
Vin de Silva and Robert Ghrist. Coverage in
sensor networks via persistent homology.
Algebr. Geom. Topol.
, 7:339–358, 2007.

[dSG07b]
Vin de Silva and Robert Ghrist. Homological
sensor networks.
Notices Amer. Math. Soc.
, 54(1):10–17, 2007.

[Ghr01]
Robert Ghrist. Conﬁguration spaces and braid
groups on graphs in robotics. In
Knots, braids, and mapping class
groups—papers
dedicated to Joan S. Birman (New York, 1998)
, volume 24 of
AMS/IP Stud. Adv. Math.
, pages 29–40. Amer. Math. Soc., Providence,
RI, 2001,
arXiv:math.GT/9905023
.

[Ghr06]
Robert Ghrist. Braids and differential equations. In
International
Congress of Mathematicians. Vol. III
, pages 1–26. Eur. Math. Soc., Zürich,
2006.

[GV]
R. Ghrist and R. C. Vandervorst.
Scalar parabolic PDE’s and braids,
arXiv:math.DS/0403308
.

[Lat91]
Jean-Claude Latombe.
Robot Motion Planning
, volume 124 of
The
Springer International Series in Engineering and
Computer Science
. Springer-Verlag, 1991.

[Sha97]
Micha Sharir. Algorithmic motion planning. In
Handbook of
discrete and computational geometry
, CRC Press Ser. Discrete Math. Appl., pages
733–754. CRC, Boca Raton, FL, 1997.