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
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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