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In my splintered readings, I have come to understand that algebraic varieties/ideals and their investigations/extensions/unifications dominated a large part of 20th century mathematics. Many profound results were achieved and prized awards given out for discoveries in this field.

But as an engineer it still escapes me why they are the quantities of central interest and how they fit into the big picture (i.e. how they provide connections between different fields of mathematics, which I think is why they are so widely investigated?). In my case I think the main issue is the vast terminology that one has to internalize before one can begin to understand even basic results. A Wikipedia reading inevitably turns to multi-hour link-fest.

To be more precise, my interest as an engineer arises specifically in their connection to dynamical systems theory. An algebraic approach to dynamical systems has been sporadically attempted since the 60s; and I think was initiated by Kalman in his study of dynamical systems over rings. Recently, far more general approaches have also been adopted incorporating category theory, sheaves etc. For example here and here.

So I am trying to find a coherent picture of their importance to (1) modern mathematics, (2) ODE's and differential geometry (3) systems theory.

Understandably, the question might be too wide to answer in a single answer, so multiple answers are invited. As for an idea of what I am looking for see this Quora answer. The answer beautifully breaks down what is probably a one line rejoinder for a mathematician into something even a sufficiently advanced high school student can understand (but the answer doesn't have to be that simple or long, intuition is more important).

I am not opposed to taking courses in Abstract Algebra to fully understand them, but from an engineering stand-point, a motivation for that type of commitment is hard to bring about unless I first get a big-picture idea of why/if it will be useful.

ITA
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  • It's unclear to me what you mean by "algebraic variety". Do you mean the set of solutions to a system of equations, as described in Dietrich Burde's answer, or do you mean the various algebraic structures usually studied in a class of abstract algebra, such as groups, ring, fields, and maybe category theory? – awkward Jul 29 '18 at 12:54
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    @awkward : given "a large part of 20th century mathematics", "many prized awards", "ideals", and "sheaves" I think it actually is quite clear which one the OP is referring to – Maxime Ramzi Jul 29 '18 at 13:05
  • @awkward, Max is right, its the latter meaning that is intended. – ITA Jul 29 '18 at 13:09
  • You might also be interested in algebraic/arithmetic dynamics (see, for instance https://math.stackexchange.com/q/187564/118539). – J W Jan 13 '20 at 12:37

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Classically, an algebraic variety is defined as the set of solutions of a system of finitely many polynomial equations over the real or complex numbers. The polynomials $f(x_1,\ldots ,x_n)$ are in $n$ variables. How can we solve polynomial equations exactly, and not just numerically? How does the solution set look like? Are there algorithms?

Algebraic Geometry and algebraic varieties in generality are of course more than this, and a survey of topics, links, and references shows why algebraic varieties are important, e.g., here:

Why study Algebraic Geometry?

Help understanding Algebraic Geometry

Dietrich Burde
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  • Thanks! Those references also seem to be a good starting point to get ones toes wet if one was inclined to start a thorough study of the subject. – ITA Jul 29 '18 at 21:29
  • The second part of Javier's answer in the first link is a great partial answer to the question depending on how much of a (non)-mathematician you are. Though heavy on terms and theorems, for someone attempting a graduate degree in EE at a fairly mathematically oriented department it was quite accessible. – ITA Jul 29 '18 at 21:52