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This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for simplicity.

Of course some of the difficulties of the present question are a subset of those of the linked question. But I think I want to be more precise here, about what's the pedagogical/heuristic obstacle I want to bypass/remove/etc, which is, after all, a detail.

So, say the engineer is happy with the start

"Algebraic geometry is the study of solutions of systems of polynomial equations in several variables..."

An engineer certainly understands this.

"...with complex coefficients..."

Here she's starting to feel a bit intimidated: why complex numbers and not just reals? But she can feel comfortable again once you tell her it's because you want to have available all the geometry there is, without hiding anything - she can think of roots of one-variable polynomials: in $(x-1)(x^2+1)$ the real solutions are not all there is etcetera.

Happy? Not happy. Because the engineer will inevitably be lead to think that what algebraic geometry consists of is fiddling around with huge systems of polynomial equations trying to actually find its solutions by hand (or by a computer), using maybe tricks that are essentially a sophisticated version of high school concepts like Ruffini's theorem, polynomial division, and various other tricks to explicitly solve systems that are explicitly solvable as you were taught in high school.

Question. How to properly convey that algebraic geometry mostly (yeah, I know, there are also computational aspects but I would contend that the bulk of the area is not about them) doesn't care at all of actually finding the solutions, and that algebraic geometers rarely find themselves busy with manipulating huge polynomial systems, let alone solving them? In other words, how to explain that AG is the study of intrinsic properties of objects described by polynomial systems, without seeming too abstract and far away?

Also, how would you convey that AG's objects are only locally described (or rather, in the light of the previous point, describable) by those polynomial systems in several variables?

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    $\begingroup$ To start, I might tell them that algebraic geometry lets us say qualitative things about solutions to equations even if it's intractable to find them quantitatively. For example, the fact that for curves, if it's genus 0 it has a dense set of rational points, genus 1 has f.g. set of rational points, and genus 2 has a finite number of rational points -- that's more arithmetic, but the point stands. I might compare this to the way dynamics can say qualitative things about solutions to differential equations, even when they can't be quantitatively solved, which might be more in their wheelhouse $\endgroup$ – Kevin Casto Oct 22 '18 at 21:55
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    $\begingroup$ One should at least try to give some geometric motivation (in other words, why it is called "geometry") and the fruitful interplay between algebra and geometry. $\endgroup$ – François Brunault Oct 22 '18 at 21:59
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    $\begingroup$ Ask the engineer questions. Find out what they really want to know about it. If they want intellectual stimulation, given them a problem that is easy to solve with AG, and a similar one that is not. If they want applications, tell them about motion planning in robotics, systems in economics, and other things that show its actual as well as potential uses. Gerhard "Makes Explaining To Myself Easy" Paseman, 2018.10.22. $\endgroup$ – Gerhard Paseman Oct 22 '18 at 22:40
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    $\begingroup$ Why do you think that algebraic geometers don't try to find the solutions? Maybe some (maybe most) algebraic geometers don't. But there are really a lot of algebraic geometers who work on computational algebraic geometry, and study solution sets. Names here include Bernd Sturmfels, David Cox, Elizabeth Allman, Andrew Sommese, Seth Sullivant, Jonathan Hauenstein... Saying algebraic geometry "doesn't care at all", even with "mostly" there, just seems incorrect, sorry. $\endgroup$ – Zach Teitler Oct 22 '18 at 23:04
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    $\begingroup$ @ArunDebray With all respect to Mumford, his great expertise in the practice of algebraic geometry did not carry over to expertise in the explanation of algebraic geometry to non-mathematicians. It seems like his idea of a non-mathematician is someone who has at the very least a college degree or graduate degree in math or physics, but may not professionally practice math. $\endgroup$ – Somatic Custard Oct 22 '18 at 23:25
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Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exist (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study of elliptic/abelian integrals: you can go from addition laws for integrals to addition laws on the curve/Jacobian...

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Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of algebraic equations.

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  • $\begingroup$ I love this book. (+1) I like it even better than Harris's and Reid's introductory books, which are already pretty geometric. $\endgroup$ – Squid with Black Bean Sauce Oct 28 '18 at 2:26
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In conversations like this, I usually lead with a concrete example of a hard problem. Complete intersections seem to work well: Observe that in general, two surfaces in three-space meet in a curve, and then ask whether, given an (algebraically defined) curve, it's always the intersection of two (algebraically defined) surfaces. How do you recognize those that are and those that aren't? This gives you a chance to talk about the value of bringing both geometric intuition and algebraic computations to the table.

Now generalize to higher dimensions. Now (if they seem to want more) you can talk about subtleties like the distinction between a true complete intersection and a set-theoretic complete intersection. Or give a sequence of increasingly challenging specific cases. Et cetera.

I've also --- though this is sort of cheating --- used the example of classifying vector bundles. This is easy to explain in the topological case: You've got, say, a circle and you want to attach a line at every point in a continuous way. You can make a cylinder, or you can make a Mobius strip. What else can you make? When do you want to consider two of these things "the same"? Now observe that the answers to these questions depend partly on the rules for how you're going to build your objects in the first place and the rules for when you consider two to be the same. If the rules are that everything has to be continuous, you're doing topology; if the rules are that everything has to be algebraic, you're doing algebraic geometry. Mention Quillen-Suslin: If the base space is itself a vector space, it's pretty easy to see that all vector bundles of a given rank are topologically equivalent, but quite hard to see the same thing in the algebraic case. Et cetera.

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If you're just trying to communicate what algebraic geometry is, without trying to convince the engineer that it's worth studying, then one simple starting point is to recall the classification of conic sections (ellipse, parabola, hyperbola) and say that one thing algebraic geometers try to do is classify the different possibilities that can occur with larger degree/number of equations/number of variables.

Notice, by the way, that there is a similar "disconnect" between engineers and mathematicians when it comes to PDEs. Engineers often just want to solve PDEs. Mathematicians are interested in solving PDEs too but are also interested in other questions. The concept of wanting to understand the qualitative features of a solution may be easier to explain in the context of PDEs, and then you can say that the situation in algebraic geometry is analogous.

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This is along the lines suggested by @DonuArapura: "describe a problem [...] that is reasonably concrete and accessible, and go from there."

Here is a problem an engineer would appreciate: Which bent pieces of wire can pass through a pinhole in a plane via rigid motions? Such curves have been called threadable curves.1

Deciding whether a given planar algebraic curve $C$ is threadable depends on the number of bitangents. For a curve of degree $d$, this number is $O(d^4)$, a result of Schubert. See the MO question, Number of bitangents to connected algebraic curve.


         



1J.O'Rourke and Emmely Rogers, "Threadable curves," Proc. 30th Canad. Conf. Comput. Geom., Aug 2018, 328—333. (arXiv abstract).

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I think a good explanation should give an idea of how algebraic geometry can make precise the idea of a generic point on an irreducible variety.

One example I might try is Gerstenhaber's Theorem, that the variety of pairs of commuting complex matrices is irreducible; a 'generic pair' is two commuting diagonalizable matrices with distinct eigenvalues. I think one could give a good idea of this without (explicitly) using the group action or topology. The engineer might already know that rotations of $\mathbb{R}^3$ are diagonalizable over $\mathbb{C}$: if so I'd explain that these matrices are still not quite 'generic' because of the $1$ eigenvalue.

If they want more, I'd go on to say that for larger number of matrices, the variety is usually reducible, so there is no reasonable idea of a 'generic tuple'.

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I would start by showing them to how find rational points on a conic. If you have a rational point then you can draw lines and find more. They will be comfortable with the geometric aspect and then you could stress the “rationality” part of the construction ie “look, the slope and the y-intercept are rational, so if one point of intersection is rational then the other one is too”

This construction has enough but not overwhelmingly many logical steps which the engineer will be able to verify should they want to E.g. rationality, getting all of them, the necessity of finding a point to start off the process.

Then you could go up to a quadratic extension (!) to “see what happens” and let them play around

I like this example because the algebra and the geometry are both at the level your audience should be comfortable with.

I wouldn’t even go to elliptic curves & the group law, in my experience it takes more mathematical exposure to appreciate those phenomena

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I have had this conversation a few times. I warm them up by introducing the concept of abstract classification of objects: I explain how a mechanical arm which can rotate in a circle, and which has another mechanical arm at the end which can also rotate in a circle, is in some sense the same thing as a torus, despite their apparent differences. The question is, then, how we can classify things that are defined by constraints that appear to be different but in some underlying way are the same?

I then explain that in A. G. the constraints are typically polynomials and this allows the use of ideas and points of view that completely general constraints don't.

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There is an answer by D. Mumford to biologists, valid also for engineers : Can one explain schemes to biologists ?

htt://www.dam.brown.edu/people/mumford/blog/2014Grothendiec.html,2014

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    $\begingroup$ One of the comments to the original question mentions this post by Mumford. $\endgroup$ – KConrad Apr 8 at 11:27
  • $\begingroup$ I am sorry. I did not know before. I found the reference in Classics Revisited : Eléments de Géométrie Algébrique, by U. Görtz $\endgroup$ – Al-Amrani Apr 10 at 13:53

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