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By applied algebraic geometry, I don't mean applications of algebraic geometry to pure mathematics or super-pure theoretical physics. Not number theory, representation theory, algebraic topology,differential geometry, string theory etc. However, since the border between pure math and applied math is kind of vague, if you are uncertain whether your answer is really about the applied algebraic geometry (for example, you know a direction that is half-pure and half-applied), please don't hesitate to add you answer or leave a comment.

To be specific, I wonder if the modern theory of schemes (and coherent sheaves, if applicable) has any applications outside of pure math while the classical theory of varieties won't be sufficient. I know there is an area called statistical algebraic geometry, but I think so far it still only uses classical algebraic geometry (no need to know schemes and sheaves). I hope to find an applied algebraic geometry area in which a background as strong as finishing most of the Hartshorne's exercises is not wasted.

Updates:

As per some of the comments below, I want to clarify some points:

(1) By "...while the classical theory of varieties won't be sufficient", you don't have to demonstrated that the research work in applied algebraic geometry you have in mind (that involves modern algebraic geometry notions) can't be translated to classical languages. I think as long as the author chooses modern language to write an applied algebraic geometry paper, there should be a reason behind it and we will find out why.

(2) As for "applicable" vs "potentially-applicable", I think nowadays it is clear many (if not most) applied math and statistics papers are only "potentially-applicable" for the time being (Just look at SIAM journals and conferences). Hence, I think "potentially-applicable" answers are welcome. If your answer is "too pure to be even potentially-applicable", someone would leave a comment below...

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    $\begingroup$ I'm skeptical that you'll get any good answers to this, primarily for sociological reasons. Applied mathematics is driven by practitioners and domain experts, not theorists, and for obvious reasons they organize their subject around things they understand rather than things that they don't. Hartshorne is a pretty serious and speculative investment for someone working on disease dynamics or fluid flow. On the other side, algebraic geometers are mostly interested in, well, algebraic geometry problems. So applications of the sort you describe would require a weird historical accident. $\endgroup$
    – Paul Siegel
    Commented May 25, 2018 at 18:11
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    $\begingroup$ That said, a lot of number theorists study cryptography, so that's not an unreasonable place to look for such an accident. $\endgroup$
    – Paul Siegel
    Commented May 25, 2018 at 18:12
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    $\begingroup$ @PaulSiegel Good comment! But I am still praying that this question will enlighten some mathematicians and trigger such "accidents". We can wait. $\endgroup$
    – No One
    Commented May 25, 2018 at 18:20
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    $\begingroup$ Does this count? arxiv.org/pdf/1303.3255.pdf specifically p.159 and p.168 $\endgroup$
    – mmm
    Commented May 25, 2018 at 19:24
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    $\begingroup$ I won't post this as an answer since I am not super-familiar with his research, but my colleague Andrew Sommese started off life as a very pure algebraic geometer (working in the post-Grothendieck context of schemes and what not) and later shifted into applications of algebraic geometry. You might check out his work: www3.nd.edu/~sommese $\endgroup$
    – Andy Putman
    Commented May 25, 2018 at 20:20

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I will ignore the issue of what is "applicable" and what is only "potentially applicable", and the issue of whether something could be translated into classical language, and simply offer an example that I came across recently:

Max Lieblich, Lucas Van Meter: Two Hilbert Schemes in Computer Vision.

As the title suggests, this is a paper on the geometry of computer vision that uses Hilbert schemes, not to mention Artin stacks.( But really it's not as fancy as all that: in the end they turn out just to be algebraic spaces.)

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Bernd Sturmfels has a large body of work applying algebraic geometry to various fields: biology, chemistry, data analysis, and computer vision.

Not all of his papers use the language of schemes, but I imagine many are at least informed by that point-of-view.

However, a quick search does reveal some papers that use such language:

  1. Algebraic Systems Biology: A Case Study for the Wnt Pathway by Elizabeth Gross, Heather A. Harrington, Zvi Rosen, Bernd Sturmfels.
  2. A Hilbert Scheme in Computer Vision by Chris Aholt, Bernd Sturmfels, Rekha Thomas.
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I am familiar with a lot of modern research in algebraic geometry (over complex or real numbers) where only very classical algebraic geometry is used (pre Grothendieck). It is both "pure" and "applied". (S. Abhyankar worked in the same department where I am:-)

But of course I cannot EXCLUDE that some modern algebraic geometry is useful in some applied questions, even very applied ones like control theory.

Usually this is a question of training of the writer, and his/her intended audience. In most cases, modern algebraic geometry (Hartshorne-like) can be translated into completely classical terms. So in many cases this is simply a choice of language and thus depends on the author's preferences. Of course, there is a problem that it is hard for people without this modern training to understand the papers written in the modern language. But there are several areas of algebraic geometry (both pure and applied) where the classical language still dominates.

As an example, where modern language is used in "applications" (to differential equations) I can mention this book:

MR1117227 Malgrange, B. Équations différentielles à coefficients polynomiaux. Birkhäuser Boston, Inc., Boston, MA, 1991.

which most people with classical training in differential equations cannot read.

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  • $\begingroup$ I partially agree with you. But a my question indicates, I want a direction/approach in which "the classical theory of varieties won't be sufficient." Do you know any applied algebraic geometry paper written in modern language that "it is hard for people without this modern training to understand"? A few references would be greatly appreciated! $\endgroup$
    – No One
    Commented May 25, 2018 at 18:13
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    $\begingroup$ I know such papers but cannot prove that "classical theory won't be sufficient". As I said, this was the author's choice. And I predict that you will not obtain an example which you want. Even if a paper is written in the modern language, no one can prove that this cannot be translated into a classical language. $\endgroup$
    – Alexandre Eremenko
    Commented May 25, 2018 at 18:16
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    $\begingroup$ You don't have to prove that "classical theory won't be sufficient" for these papers at all (no one can)... please just add those papers in your answer... $\endgroup$
    – No One
    Commented May 25, 2018 at 18:19
  • $\begingroup$ Are the examples that you have truly "applied" rather than possibly "applicable"? For instance, I've asked experts in robotics if they ever use any geometry in their work, and they usually respond that they remember hearing about it in graduate school but that the problems it solves are at best peripheral to the subject. They do not say the same about, say, statistics or linear algebra. $\endgroup$
    – Paul Siegel
    Commented May 25, 2018 at 18:23
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    $\begingroup$ Yes, some sophisticated (real) algebraic geometry is used in robotics. People who do this are unfamiliar with the modern language. They use classical language. $\endgroup$
    – Alexandre Eremenko
    Commented May 25, 2018 at 18:29
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There is a topic that could be called "phylogenetic algebraic geometry"; see for example On phylogenetic trees - a geometer's view by Buczyńska and Wiśniewski. This work is arguably only "potentially applicable," and the use of modern scheme-theoretic language is arguably not really necessary, but maybe it is close to what you are looking for.

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Related to several answers is the issue of identifiablity and equations for secant varieties. This has actual and real uses. Identifiable means that a point in a secant variety, or at least a general point of the variety has a unique representation. It's related to something called Waring's problem for polynomials. This comes up in phone communication. I believe that the way data is transmitted over channels is that all the information is put in a numeric form, raised to a high power and then summed. The identifiability allows one to find the original numbers, which is the information required. So I've been told.

People have mentioned phylogenetics. The models of phylogenetics are secant varieties to Segre/Veronese embeddings of projective spaces. The equations of these varieties are the information of the model. Shmuel Friedland won a smoked Copper River Salmon for finding the (set theoretic) equations of a specific model of interest to a specific person. In addition, for phylogenetics and transmission, having the actual equations of the vareity is somewhere between useful and necessary as checking (approximate) equality is much easier with equations of lower degree.

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    $\begingroup$ Adam, I can witness it was only half of a salmon-- since the prize giver decided it was only half a solution... $\endgroup$
    – Margaret Friedland
    Commented May 31, 2018 at 23:57
  • $\begingroup$ @ Margaret - I will ask the author and the eater- perhaps because he only gave a set theoretic solution ? $\endgroup$
    – meh
    Commented Jun 3, 2018 at 22:10
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Maybe research at the intersection of algebraic geometry, representation theory, and deep learning might be of interest to you.

Double framed moduli spaces of quiver representationshttps://arxiv.org/abs/2109.14589

This builds on previous work on the representation theory of neural networks, giving category-theoretic formulations too!

This one also has the same flavor; authors define a Kähler metric to perform gradient descent on the moduli space of quivers. Kähler Geometry of Quiver Varieties and Machine Learning

For an even more abstract/categorical approach. Try this by Yuri Manin and Matilde Marcolli. It's Algebraic topology and category theory, not algebraic geometry though.

HOMOTOPY THEORETIC AND CATEGORICAL MODELS OF NEURAL INFORMATION NETWORKS

Finally, the paper below cites the work by all of the authors above. But it bridges the abstract approach with the more concrete goal of developing better optimization techniques of neural networks derived from the geometry and symmetries of their parameter spaces.

THE QR DECOMPOSITION FOR RADIAL NEURAL NETWORKS

The following quote of the authors encapsulates their more pragmatic goal.

"Manin and Marcolli advance the study of neural networks using homotopy theory, and the “partly oriented graphs” appearing in their work are generalizations of quivers. In comparison to the two aforementioned works, our approach is inspired by similar algebro-geometric and categorical perspectives, but our emphasis is on practical consequences for optimization techniques at the core of machine learning."

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