I am currently pursuing an undergraduate degree in Electronics and Communications Engineering. I recently got a research internship to study algebraic geometry for two months. I would like to know what exactly the applications of algebraic geometry are, in the field of electronics and communications, signal processing, control theory and other areas. In short, I'm curious to know about the real-world applications of algebraic geometry.
[Math] What are the applications of algebraic geometry to electronics
algebraic-geometryapplications
Related Solutions
It's a massive subject, and there are many different perspectives; here are a few that don't require too much background.
Perspective one: It's a generalization of linear algebra.
Linear algebra is about dealing with systems of linear equations. This is easy: the set of solutions to a (homogeneous) system is just some subspace of $F^n$ (where $F$ is the field of scalars), and you can compute its dimension by row-reducing your system into echelon form.
Algebraic geometry is about dealing with systems of polynomial equations. As you may imagine, this is much harder. In linear algebra, much of the theory is entirely independent of the field $F$, at least until you want to diagonalize operators; in algebraic geometry, non-algebraically-closed $F$ are a massive headache, and there are phenomena in characteristic $p$ that don't show up in characteristic $0$.
Perspective two: It's a computational tool in classical geometry.
In geometry and topology we may wish to study invariants of manifolds. We define lots of invariants, e.g., homology groups, but how can we get our hands on them? For most examples, we can't do it easily at all, but if the example happens to be a complex manifold given by polynomial equations, there's a lot more that we can say. This is especially important if you want to do things with computers.
Perspective three: It's a conceptual way to think about commutative algebra.
If I give you some ring, OK, great, it has prime ideals, maximal ideals, zero divisors, etc. What does all this mean, and how do you ever remember the barrage of technical theorems about integrality, Artin rings, regular local rings, etc?
If the ring is the ring of functions on some space, then the geometry of the space may reflect properties of the ring, and we can remember the commutative algebra by picturing the geometry. What Grothendieck realized is that if we define "space" correctly (which is not so easy), every ring is the ring of functions on some space! For an example of how you might relate geometry to intrinsic properties of the ring: the space attached to a ring is connected if and only if all of the zero divisors in the ring are nilpotent.
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).
Best Answer
It is difficult to say anything not too generic but the very typical problems that appear in engineering are:
1) Can I decompose this complicated problem into easier to understand problems? 2) I understand simple relationships. Can I extrapolate these simple relationships into solutions of more complicated systems?
You've likely seen 1) as an electrical engineer. You always try to take complicated functions, such as a triangle wave, which are hard to create in nature and use your basic functions like sine and cosine to reconstruct them.
Maxwell's equations are an example of 2) but even more fundamental is Newtonian mechanics. It is easier to understand that velocity is linear than to say that a falling object has position given by a parabola. It is easier to understand that the time-change in a current in an inductor induces a proportional change in voltage than to see that the voltage is an exponential.
Back to your original question, what does this have to do with algebraic geometry? It turns out that many of the questions asked in these contexts a) extend to the complex numbers b) extend to projective spaces.
With appropriate adjectives inserted, analytic questions on projective spaces over the complex numbers can be phrased as algebraic questions via Serre's GAGA theorem. Writing the triangle wave as a sum of sine and cosine functions, while on the surface looks strictly analytic, actually can be deduced by working with algebraic functions on the projective complex line.