Straight forward question for someone who’s upper most math class was calc 1 a long time ago. Also bonus points if someone can explain why Grothendieck is the “Einstein of math”
In: 7
Algebraic Geometry is interpreting algebraic questions geometrically, and then use this to gain new insights. The formalism is quite involved, using the language of sheaves and sites among others. But let me give three elaborate examples of increasing depth, but hide some technicalities:
(1) Assume you want to find all solutions S of, say x² + y² = 1. One solution is x=-1, y=0. Now to find another one, we try the following geometric(!) idea: consider any line L through this point (-1,0) and through any fixed point (0,t) on the y-axis. Then a new solution to the initial problem can be found as the intersection point of L with S.
As an equation, this line is calculated to be given as y = t·(x+1) for some t. So we can substitute this into x² + y² = 1, solve for x and get x = (1-t²)/(1+t²), then get y = 2t/(1+t²). [Picture from Wikipedia to make this easier to follow](https://upload.wikimedia.org/wikipedia/commons/a/a5/Rational_parameterization_of_unit_circle.svg).
Conclusion: we get solutions by choosing any t and then set s,y as above. Conversely, any solution x,y defines a line L through (-1,0) and (x,y), which shows that we actually got all of them this way!
Now the really interesting observation is that we never used “where” our numbers come from, if we want real solutions (aka a “circle”), then we let t wander through the real numbers and get a new decription of a circle. But equally, we can restrict t to rational numbers to get exactly the rational solutions! This is (after clearing denominators) exactly the answer to the much less obvious [Pythagorean triples](https://en.wikipedia.org/wiki/Pythagorean_triple#Rational_points_on_a_unit_circle).
(2) Elliptic curves E, essentially the solutions of equations of the form y² = x³ + ax +b (there are some caveats here, but lets skip those). They for example appear in cryptography, so they are even relevant in our modern real life (e.g. banking). A nice geometric fact is that the complex solutions form exacty a torus, a.k.a. the surface of a donut. This has quite a few implications on the possible solutions.
Furthermore, you can define an “addition” on the solutions in a geometric way: if P=(x,y) and Q=(z,w) are two solutions, draw a line L through those two points, and check that there is exactly one more point where L and the elliptic curve E intersect [(Picture)](https://www.researchgate.net/profile/Tabassum-Ara-2/publication/326009351/figure/fig1/AS:642029855977473@1530083254406/Point-Addition-on-the-Elliptic-Curve-18.png). Mirror this point at the x-axis to get a new point R and call it the “sum” of P and Q.
This addition is then useful to define cryptographic protocolls, but that would go even farther to explain.
(3) Curves of “genus at least 2”: assume that you have some equations which geometrically describe some prezel-like thing with at least two holes [(Picture)](https://upload.wikimedia.org/wikipedia/commons/b/bc/Double_torus_illustration.png). Then a result by Faltings says that this geometric info is enough to conclude that there are only finitely many solutions within the rational numbers. This e.g. applies directly to Fermat’s Last Theorem, asking for the solutions of x^n + y^n = z^n if n>2.
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