[Robert Ghrist](http://www.math.upenn.edu/~ghrist/index_files/research.htm)'s web site has some interesting notes on robotics and algebraic topology:

> My work focuses on those methods in applied mathematics which are topological in nature. Such methods have the feature of being very robust: topological results are tolerant of the "noise" inherent in physical systems. Such techniques are therefore both elegant and effective in engineering and science.

I first came across him reading in the *Notices* about the theory of [**barcodes**](http://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf) and [**persistent homology**](http://ftp.cs.duke.edu/~edels/Papers/2008-B-02-PersistentHomology.pdf).  

Currently, a lot of tools only use a 1D graph theory based approach.  Using the Rips complex, a tool from geometric group theory, shows how to find topological features in data sets made of discrete points.     Somehow we have to "complete" the point-set into a topological space.

![enter image description here][1]

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There is an even more direct example of Grothendieck's influences in the theory of sensor networks.  Please look at Michael Curry's [Sheaves, Co-sheaves and Applications](http://arxiv.org/abs/1303.3255v1)  which has a short introduction to the history of sheaves and their generalizations, mentioning Leray, Grothentieck, Kashiwara, MacPherson and others.

> In 2008, Robert Ghrist initiated a call to bring sheaf theory, specifically sheaf cohomology, to bear on a variety of applied problems.  *Euler calculus* - a decategorification of contructible sheaf theory, has already made progress twoard this goal.  Heuristically, sheaf cohomology would provide calculable summary of the topology of data and programs, even if initially there is no topology in sight.  

The idea is that sheaves help organized information spread out in different places into a single global piece of information.

  [1]: https://i.sstatic.net/4MsQu.png