“Real”-life applications of algebraic geometry

Before you tell me that this question has been asked, give me a bit of your time please to read this question because it is not as simple as it sounds.

I did my undergraduate degree in mathematics, taking a pretty heavy course load in theoretical math and doing really well in it. I decided not to proceed with math and am continuing on to a professional degree.

However, every once in a while I have doubts about my decision because it was made on more than one basis, i.e. passion for a given academic subject. One of the reasons was that, seeing my professors, it seemed that mathematicians were very much living in a world of their own and every time I asked for an application of what I was studying, e.g. Galois theory, I got two sorts of answers:

1. An application of the concept in another area of mathematics...which is not what I was looking for.

2. A trivial application where a physical/computational/etc system is "modelled" with the concept, e.g. something is a "group", but the recognition that it is was completely useless since the application did not produce a result that would have been otherwise unknown.

My question is: If I changed my mind, applied to do a graduate degree in mathematics and decided to work in a field outside of academia, would I have useful applications of what I studied (and not just a tiny fraction of what I studied, e.g. ODEs) in "real" life"?

I'm very much interested in algebraic geometry (and I am being honest when I say that it is one of the rare things that makes me truly giddy thinking about it). I think you would really answer my question if you could give me an example of a real life problem (not in excessive detail) that was solved thanks to techniques of algebraic geometry. I don't think I have the knowledge of AG to understand the details, so I am more interested in the statement of the "real"-life problem and the non-trivial result of its mathematical modelling using concepts in algebraic geometry.

Thank you so much! I really appreciate your help. I'm really trying to do some soul searching here and you could really help me with it.

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When you say "application", do you mean something utilitarian? For instance, I would say the primary application of Galois theory is simply a deeper understanding of how numbers work, and in my opinion that's enough. If you really want to solve quintics for some practical application, just plug them into a computer. – Jack M Feb 16 '14 at 14:22
That sounds like what the OP referred to as an application to mathematics, @JackM. – Alexander Gruber Apr 23 '14 at 5:43

There is some effort afoot to derive some use of algebraic geometry for geometric computer vision (multiple view geometry). Specifically groebner basis. There are some polynomial systems of equation associated with multiple view geometry and certain attemps use groebner basis to get simple formulas for special cases as a 'preprocessing' step. So the algorithms that actually work on images just have hard-coded equations rather than doing symbolic math such as groebner basis during runtime.

For more on this see these links:

minimal problems

A Hilbert Scheme in Computer Vision

A QCQP Approach to Triangulation

However, be aware that this is cutting edge research and still in the business of establishing itself. It is almost silly how many different methodologies there are for these and other problems in computer vision. The jury is still out - any one of those may prove itself as the most pertinent in the future.

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Algebraic Geometry has applications in Cryptography. See for instance these links:

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Here's an example of a real-life'' application of algebraic geometry. Consider an optimal control problem that adheres to the Karush-Kuhn-Tucker criteria and is completely polynomial in nature (being completely polynomial is not absolutely necessary to find solutions, but it is to find a global solution).

One can then use the techniques of numerical algebraic geometry (namely homotopy continuation) to solve this system of (nonlinear) polynomial system, find all the complex solutions, throw out any that have too large'' of an imaginary part, attain all the real solutions, and check for the optimal one.

A number of software packages exist that can do this (HOMPACK, Phcpack, HOM4PS2.0, POLYSYS_GLP, POLYSYS_PLP).

Some other real-world applications include (but are not limited to) biochemical reaction networks and robotics / kinematics.

These ideas start with Davidenko (50's) and then greatly improved independently by (Drexler) and (Garcia and Zangwill) (late 70's).

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I can't give you a real life problem, but I know people are using algebraic geometry at Sandia National Lab. I heard a Professor talking about it once. Some of the those government laboratories might be a great place to look!

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If interested, I can probably put you in contact with that Professor! – user66360 Nov 20 '13 at 21:35
Dear @kbbal, this qualifies more as a comment. – Ehsan M. Kermani Nov 20 '13 at 23:22