Is algebraic geometry constructive?

Question

Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an informal manner.

I hope that this question belongs on this site, since it is not strictly research-level.

As an autodidact (I have an ongoing formal education in physics, but the amount of math we learn here is abysmal, so most of my mathematics knowledge is self-taught), I have noted that algebraic geometry seems to be really impenetrable for somebody who has no formal education in the field, unlike, say, differential geometry or functional analysis, which are areas I can effectively learn on my own.

Pretty much every time I encounter AG-related stuff on sites such as this one or math.se, I see layers and layers of abstractation on top of one another to the point where it makes me wonder, is this field of mathematics constructive, in the sense that can it be used to actually calculate anything or have any use outside mathematics?

The point I am trying to make is that, using differential geometry as an example, is is constructive. No matter how abstractly do I define manifolds, tensor fields, differential forms, connections, etc, they are always resoluble into component functions in some local trivializations, with which one can actually calculate stuff.

Every time I used DG to calculate stellar equilibrium or cosmological evolution or geodesics of some model spacetime, I get actual, direct, palpable, realizable results in terms of real numbers.

I can use differential forms to calculate the volumes of geometric shapes, and every time I use a Lagrangian or Hamiltonian formalism to calculate trajectories for classical mechanical systems, I make use of differential geometry to obtain palpable results.

On top of that, I know that DG is useful outside physics too, I have heard of uses in economy, music theory etc.

I am curious if there is any real-world application of AG where one can use AG to obtain palpable results. I am not curious (for the purpose of this question) about uses to mathematics itself, I know they are numerous. But every time I try to read about AG I get lost in the infinitude of sheaves, stacks, schemes, functors and other highly abstract objects, which often seem so impossible to me to be resolved into calculable numbers.

The final point is, I would like to hear about some interesting applications of algebraic geometry outside mathematics, if there are any.

Answer 1

If you forget about all the layers of abstraction, algebraic geometry is, ultimately (and very roughly speaking), the study of polynomial equations in several variables, and of the geometric objects they define. So in a certain sense, whenever you're doing anything with multivariate polynomials, there's probably some algebraic geometry behind it; and conversely, algebraic geometry questions can generally be reduced, at least in principle, to "does this system of polynomial equations have a solution?". Now to go a (little) bit further, one should consider the field upon which these equations are defined, and more importantly, in which the solutions are sought. The whole point of algebraic geometry is that most of the formalism can be made uniform in the base field, or indeed, ring. But there are at least two main flavors:

• Geometric questions are over algebraically closed fields (and all that really matters, then, is the characteristic of the field). These questions are, in principle, algorithmically decidable (although the complexity can be very bad), at least if we bound every degree involved in the problem; Gröbner bases are a the key tool to solve these geometric problems in practice.

• Arithmetic questions are over any other field, typically the rational numbers (→ diophantine equations); for example, an arithmetic question could be "does the curve $x^n + y^n = z^n$ (where $x,y,z$ are homogeneous coordinates) have rational solutions beyond the obvious ones?". Arithmetic questions can be undecidable, so there is no universal tool like Gröbner bases to solve them. There is a subtle interplay between geometry and arithmetic (for example, in the simplest nontrivial case, that of curves, the fundamental geometric invariant, the genus $g$, determines very different behaviors on its rational points according as $g=0$, $g=1$ or $g\geq 2$).

Then there are some fields which are "not too far" from being algebraically closed, like the reals, the finite fields, and the $p$-adics. Here, it is still decidable in principle whether a system of polynomial equations has a solution, but the complexity is even worse than for algebraically closed fields (for finite fields, there is the obvious algorithm consisting of trying possible value). Some theory can help bring it down to a manageable level.

As for applications outside mathematics, they mostly fall in this "not too far from algebraically closed" region:

• Algebraic geometry over the reals has applications in robotics, algebraic statistics (which is part of mathematics but itself has applications to a wide variety of sciences), and computer graphics, for example.

• Algebraic geometry over finite fields has applications in cryptography (and perhaps more generally boolean circuits) and the construction of error-correcting codes.

But I would like to emphasize that the notion of "applications" is not quite clear-cut. Part of classical algebraic geometry is the theory of elimination (i.e., essentially given a system of polynomial equations in $n+k$ variables, find the equations in the $n$ first variables defining whether there exists a solution in the $k$ last): this is a very useful computational tool in a huge range of situations where polynomials or polynomial equations play any kind of rôle. For example, a number of years ago, I did some basic computations on the Kerr metric in general relativity (ultimately to produce such videos as this one): the computations themselves were differential-geometric in nature (and not at all sophisticated), but by remembering that, in the right coordinate system, everything is an algebraic function, and by using some elimination theory, I was able to considerably simplify some symbolic manipulations in those computations. I wouldn't call it an application of algebraic geometry to physics, but knowing algebraic geometry definitely help me not make a mess of the computations.

Answer 2

Yes it is! We have Gröbner basis algorithms that can answer the question of ideal membership and can be used to answer many other geometric questions. If you are interested in this further, Cox, Little O'Shea have a good introduction to algebraic geometry from this perspective.

Answer 3

Perhaps you will not consider this as a real-world application, but in recent years Theoretical Computer Science is using more and more Algebraic Geometry. For example, one main approach for attacking the "P vs NP" problem is based on Algebraic Geometry (see this Wikipedia page). There are various algorithms that rely on tools from algebraic geometry (see for example this paper). Other uses of Algebraic Geometry pop up in Cryptography, Coding Theory, and other sub-fields.

Answer 4

Slightly roughly, a smooth algebraic variety over $\mathbb{R}$ has an open covering by algebraic varieties of the form $U = \{ (x_1 , \ldots , x_n) \in \mathbb{R}^n \ | \ f_1 (x_1 , \ldots , x_n) = 0 , \ldots , f_m (x_1 , \ldots , x_n) = 0 \}$ where the $f_1 , \ldots , f_m$ are polynomials (with some condition on derivatives making this closed submanifold of $\mathbb{R}^n$ smooth). One can then describe the transition function and cocycle condition etc., so this is exactly like a smooth manifold, but in fact of a more restrictive nature (you can only construct from manifolds which are cut out by polynomials).

You can then cover $U$ by open subsets of $\mathbb{R}^{n - m}$, but this will be "not algebraic". An algebraic way of doing this further simplification is called "etale cover".

One then needs a series of Q&A's, to explain why you then see mountains of abstraction, so perhaps this is not the place. Basically and roughly, this is because you can ask for solutions of the same polynomial equations also over $\mathbb{C}$ and over $\mathbb{Z} / p \mathbb{Z}$ and in fact over a lot of other fields and rings, and it turns out that even if you were interested in only one ring, the "manifolds" that you get from others turn out to be related and interconnected, and contain useful information for your original pursuit. Thus, one would like to think of an algebraic manifold as being over all rings at once, and then there are various devices and theories which look quite abstract if one is not used to them, for dealing with this.