# “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.

• 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

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).

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

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!

• 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

I finished my PhD last year and it was on the kinematic analysis of some special robots. I used computational algebraic geomtery (CAG) to analyse them. One of the examples that I can give you is the kinematic analysis of a simple four-bar linkage with equal link lengths. Concepts from algebraic geometry (such as prime decomposition of ideals) tells us that this mechanism can have three different kinds of degrees of freedom (called operation modes in the kinematics lingo): two rotations and a translation. It also tells us that the mecanism has singularities between these modes (which are nothing but the singularities of the algebraic variety corresponding to its motion). We used these singularities to build a compliant gripper that can have angular and parallel grasping capabilities. Although we built a 3d printed prototype, I believe that this gripper can be manufactured for different applications. If you are interested, you can find more details in my PhD thesis: https://uncloud.univ-nantes.fr/index.php/s/xoc9KErbyJWY55N

Also, at the moment, I am working on a control problem in robotics. For some serial robots (that resemble an arm), a camera is placed at the free end which is used to track some points (could be other gemetric features) on an object, which is then used to control the motion of the robot. This is called as vision based control. The problem is that there are some configurations in which you can't control the robot (called as singularities of the control model). For instance, if the three points on the object are collinear, the robot can rotate about that line and it'll think that it is stationary since the projections of those points on the image plane of the camera will be stationary. We recently succeeded with the help of CAG tools that if there are 4 points, there are at least 2 positions of the camera that are singular and at most 6. Thus, we could test them on a real working robot and inform the robotician to avoid those locations of the points on the object as well as that of the camera so that they could control the robot without any problems. I will update the answer with a link to our article as soon as soon as it is published.

• That's very interesting to see. I am majoring in Mechanical and minoring in mathematics and I haven't seen much of pure mathematics being used in mechanical. I am very much interested in algebraic geometry/topology/etc as of now. – Mann Nov 1 at 16:05
• Yes, especially kinematics has always attracted geometers and other mathematicians throughout history. However, there have been great developments recently to use concepts like line geometry, algebraic geometry and non-Euclidean geometry to solve problems in kinematics and other fields of robotics. – Abhilash U N Nov 3 at 20:10

I know that the (projective) algebraic geometry is used to create algorithms for the reconstruction of images and pictures.

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 would say: Probably not, no.

And if you really want the answer to be yes, then maybe you have made the right decision not to pursue graduate study in 'pure' mathematics like algebraic geometry.

To be honest though, to have found something that makes you giddy just to think about: Wow, right? Many of us academics suspect that we will not be able to find this kind of satisfaction outside of academic mathematics.

Application of algebraic geometry is popular nowadays:

http://meetings.siam.org/program.cfm?CONFCODE=AG17&_ga=2.48674598.1478910260.1545340527-1508289771.1542665618

There are certain successes in coding theory. Numerical tools are giving us an edge to be able to compute things that were not possible before. There is also a growing literature on optimization of polynomial objective functions. All of these can be used for real life problems up to their limitations - nature seems to be always more complex than the math we can compute :) -