I don’t have a mathematical background and am looking for an “intuitively satisfying” explanation (so, for example, the Wikipedia article is way too technical). Perhaps this is not possible in which case, fair enough.

I understand (I think) what a polynomial is and I believe algebraic geometry is about understanding the solutions to polynomial equations using abstract algebraic techniques and geometry. I rapidly get lost when the discussion shifts to rings, fields, schemes and so on. However, I’m not looking to understand all these different concepts but rather get a high level overview.

One day, I’d like to understand how Grothendieck revolutionized the discipline but that may be far too ambitious 🙂

In: Mathematics

There really isn’t a great way to distill this down to an ELI5 level imo, especially if you aren’t yet comfortable with polynomials

For context, polynomials are something introduced to 13 year olds, but abstract algebra courses (which provide the framework for discussing algebraic structures like groups, rings, etc) typically aren’t taught until upper division math courses

There’s a *lot* of math in between

I’ll try the eli5.

Geometry study the “shape” of things.

Algebra study the “structure” of mathematical notions.

So let’s take an exemple : a rotation is a geometric transformation that does not change the shape of an object. This is a geometrical notion.

Now we know that if we do two rotations from the same center , the outcome is the same as one rotation. (Like if you rotate 90° and then 30°, it is like rotating 120°. Or if you rotate 40° , then -40°, it is like rotating 0°).

So assuming we keep the center, a rotation combined with a rotation is a rotation. And you have a rotation that does nothing. And you can always cancel a rotation by rotating in the other direction.

This gives a lot of information about the *structure* of this set of rotation. This structure is a well known algebraic structure (in my example it is a group). But we also know A LOT about group and what’s going on in this kind of steucture, so we can use our knowledge about group structures to get knowledge about rotation. And by doing that, we did algebraic geometry.

Geometry is the study of shapes.

Algebra is reducing complex things to basic rules, then understanding the complex things in terms of those rules.

Algebraic geometry involves breaking shapes down to simpler structures and understanding/extending those shapes from those structures.

Sort of like how a square can be broken into triangles. If you understand the triangles, you can say a lot about the square.

(I should mention I’m not an expert, my exposure to algebraic geometry is limited to a course which I struggled in. I did pass though).

Ultimately _algebraic geometry_ is about the application of geometric ideas to algebraic problems. That is a very broad phrasing, but it is the gist of it.

Say you have a bunch of linear equations such as 3x + 4y + 5z = 6. Geometrically the solutions of this one form a plane in 3D, and a general linear equation gives you some arbitrary-dimensional analogue. If you solve all of them, then geometrically you are looking for the _intersection_ of those planes. Even the ancient Greeks already intersected lines and circles, so this is a very old idea.

This allows you to make multiple observations that are less easy to spot when doing just algebra. For example, two parallel planes never intersect and hence there is no solution. Even more importantly we can speak about dimension at all (we already did!). As an actual application: if the planes are not in very specific arrangements (which have probability zero to happen if chosen randomly), then we expect each subsequent equation to reduce the dimension(!) of the solutions by one. So n equations in n variables will typically have 0D solutions, actually exactly one solution (a point, which is 0D!); and n+1 equations in n variables will usually not have any solution at all.

But what I just said does not only apply to _linear_ equations. You can take quadratic, cubic, arbitrary polynomial ones and those statements are still true! A “new” relationship between variables will always decrease the solution’s dimension by exactly one, n such equations in n variables usually have finitely but non-zero many solutions.

This is barely a starting point, though. A good next step would be to look at _elliptic curves_ often written as y² = x³ + ax + b. We can by pure geometry define [“addition” on the solution points](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/ECClines.svg/680px-ECClines.svg.png) which actually allows us to classify those solutions among many things; and do cryptography with them.

Are you familiar with the concept of a “function”?

A one-variable function such as f(x), g(x), etc. (read as “f of x”, “g of x”) is a relation between input numbers and output numbers, so that each valid input produces exactly one output. So, for example, x^2 is a function, because for every possible real number x you could come up with, you’ll get exactly one output: f(3)=9, f(-2.2)=4.84, etc. Another function is f(x)=ln(x), the natural log, which does put out only one value, but isn’t defined for 0 or negative inputs. The “graph” or “plot” of any given function is all of the points where the function has a defined output.

All of the examples above are one-variable functions, but you can have a function that has more than one variable. For example, if you wanted to talk about how strong two magnets push against each other, you’d care about how strong those magnets are and how close together they are. This is usually written as something like f(x,y)=y^2 – x^3 or similar. Multivariable functions like this can have much more complicated graphs.

Algebraic geometry is mostly focused on finding solutions for these equations, usually, the places where a given point where x=M and y=N, usually written as (M,N), causes the overall function f(x,y) to evaluate to 0 (so sometimes “solutions” are called “zeroes”). Because the equations are related to the graph, applying the rules of algebra (which would be too complex to explain quickly in an ELI5) to the equations allows us to answer difficult geometric questions by manipulating the equations instead. When studied in this way, the equations are called “algebraic varieties.” There are some surprising results that can come from using the tools of algebra on these equations, but because it’s very heavy on the “algebra” side of things, it can feel like there isn’t a whole lot of *geometry* involved–it’s not really “geometry” like what you would have studied in primary and secondary school.

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