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I was reading about the passing of Alexander Grothendieck, and something caught my interest:

Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal mathematical principles that could shed unexpected light on them. Applications of his work are evident in fields as diverse as genetics, cryptography and robotics. New York Times

After extensive googling, I haven't been able to find examples. Has Grothendieck's mathematical work been applied to robotics, cryptography, or genetics, and if so, how?

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    $\begingroup$ It is fair to ask the question, since the NYT said this. But you are right to be skeptical. There are no major applications in these field though it would be foolhardy to say no one ever claimed to see connections. $\endgroup$
    – Colin McLarty
    Commented Nov 19, 2014 at 3:28
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    $\begingroup$ Please do not hastily close this question. The use of matrices in robotics is not relevant (as far as I know), but people do study configuration spaces of linkages and other models of robots. I'm not qualified to talk about them, but some people consider a Grothendieck ring of configuration spaces. See Topological Robotics, and this section: books.google.com/… $\endgroup$
    – Douglas Zare
    Commented Nov 19, 2014 at 3:36
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    $\begingroup$ I do not think that MO is well-suited to speculation about what might or might not have been in the mind of a reporter for the New York Times. $\endgroup$
    – Steven Landsburg
    Commented Nov 19, 2014 at 6:48
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    $\begingroup$ This question appears to be off-topic because it is about some tangential remark in a non-mathematical text. $\endgroup$
    – user9072
    Commented Nov 19, 2014 at 19:09
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    $\begingroup$ I don't understand why this is considered off-topic. If someone asked the exact same question but without the quote from a reporter it would be a legitimate question, would it not? It's about professional mathematics, at least one source has been mentioned that does name inventions of Grothendieck, and it's clearly answerable, even if mostly in the negative. The question is not about what the reporter thought, the question is about whether they were right, which is a mathematical question with a mathematical answer. $\endgroup$
    – Robert Mastragostino
    Commented Nov 19, 2014 at 23:24

4 Answers 4

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Here is a guess: first, "Grothendieck's work" is being interpreted as "algebraic geometry," so the real question is what applications of algebraic geometry there are in genetics, cryptography, and robotics.

  • Genetics: my guess is that this is a reference to the use of algebraic statistics to understand phylogenetics. See, for example, this news article, this survey, or this textbook.
  • Cryptography: my guess is that this is a reference to the use of elliptic curves in cryptography as well as perhaps to the use of algebraic curves to produce error-correcting codes. See, for example, this textbook.
  • Robotics: my guess is that this is a reference to the use of algebraic geometry to understand robot motion planning. See, for example, this paper.

However, as far as I can tell, Grothendieck's work in particular is not relevant to any of this.

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    $\begingroup$ Qiaochu, don't forget the importance of schemes in robotics research being done at your alma mater. newsoffice.mit.edu/2013/… $\endgroup$
    – Dylan Yott
    Commented Nov 19, 2014 at 4:31
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    $\begingroup$ @Dylan Yott: That is not a mathematical scheme, of course, but it is natural to consider a scheme structure on some configuration spaces. $\endgroup$
    – Douglas Zare
    Commented Nov 19, 2014 at 4:52
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    $\begingroup$ I used to carry around Polydrons to show the following example, which I learned from Bob Connelly: You can build a nonconvex polyhedron from triangles that is provably rigid, so the configuration space mod rigid motions of space is locally just a point. However, the model flexes in your hand. This is because the constraints give a condition like $x^4=0$ instead of $x=0$, and so even small tolerances for error produce macroscopic motions. I think this is natural to describe using schemes. $\endgroup$
    – Douglas Zare
    Commented Nov 19, 2014 at 5:06
  • $\begingroup$ @DouglasZare: Would you mind explaining how to build such configurations? That sounds pretty cool. $\endgroup$
    – Daniel Litt
    Commented Nov 21, 2014 at 23:08
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    $\begingroup$ @Daniel Litt: The configuration Bob Connelly showed me was made of $32$ regular triangles. Start with a stellated cube, and pull out one square pyramid, placing a square antiprism between the cube and the square pyramid. The apparent motion is an arc. You can do the same on the opposite side to get a two dimensional infinitesimal motion. There are simpler versions if you can fix some points. Connect two unit segments with endpoints fixed at $(0,1)$ and $(0,-1)$. Their connection must be at $(0,0)$ but the condition is $y=0, x^2=0$, giving an infinitesimal motion in the $x$ direction. $\endgroup$
    – Douglas Zare
    Commented Nov 22, 2014 at 3:37
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To expand upon Douglas Zare's comment a bit, in the book

Farber, Michael. Invitation to topological robotics. European Mathematical Society, 2008,

Farber reports on Światosław Gal's use of C&P (cut & paste) surgery and the Grothendiek ring $\mathfrak{C}$&$\mathfrak{P}$ to compute Euler characteristics of polyhedral configuration spaces (pp.55-56):

        C&P

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    $\begingroup$ It may be worth noting that in that chapter Farber is surveying work of Gal: journals.impan.pl/cgi-bin/doi?cm89-1-4 $\endgroup$
    – Mark Grant
    Commented Nov 19, 2014 at 13:23
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    $\begingroup$ The next question is how much topological robotics is applied in actual real-life robotics. $\endgroup$
    – Lennart Meier
    Commented Nov 19, 2014 at 13:55
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    $\begingroup$ Ghrist and collaborators have been applying concepts from Sheaf cohomology in robotics applications. $\endgroup$
    – Piyush Grover
    Commented Nov 19, 2014 at 14:59
  • $\begingroup$ @LennartMeier: Definitely real robotics uses configuration spaces extensively, often via sampling. E.g., "Dynamic efficient collision checking method of robot arm paths in configuration space." But I do not know how much this type of work depends on topological considerations. So: good question. $\endgroup$
    – Joseph O'Rourke
    Commented Nov 19, 2014 at 17:52
  • $\begingroup$ @MarkGrant: Thanks, Mark, incorporated your correction. $\endgroup$
    – Joseph O'Rourke
    Commented Nov 19, 2014 at 18:23
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Robert Ghrist'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 and persistent homology.

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.

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There is an even more direct example of Grothendieck's influences in the theory of sensor networks. Please look at Justin Curry's Sheaves, Co-sheaves and Applications 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 toward 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.

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    $\begingroup$ Although this is quite interesting: Where is the influence of Grothendieck here? Grothendieck certainly had important influence on algebraic topology, but not on this kind of algebraic topology. Simplicial complexes, (Cech) (co)homology etc. all existed before Grothendieck became active. $\endgroup$
    – Lennart Meier
    Commented Nov 19, 2014 at 17:37
  • $\begingroup$ @Lennart, some of recent work by Ghrist involves sheaf cohomology. Maybe thats the connection to Grothendieck ? $\endgroup$
    – Piyush Grover
    Commented Nov 19, 2014 at 18:34
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    $\begingroup$ @PiyushGrover I've now checked the article by Curry above and also math.upenn.edu/~ghrist/preprints/eulertome.pdf They seem indeed not only to use Cech cohomology, but e.g. higher direct images (and maybe even derived categories in Verdier duality). Furthermore, ideas on Grothendieck topologies and tame topology seem at least to have been an inspiration. $\endgroup$
    – Lennart Meier
    Commented Nov 22, 2014 at 15:00
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Genetic research has some applications of grothendieck's theory, above mentioned answers inform about grothendieck cuts, and their applications to robotics. Some cancer researchers use groupoids, so to determine a gene expression, define first a grothendieck space, and then study deformations. These (Yoneda-Grothendieck) structures are useful in studying the dynamics of the cancer gene.

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