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
I can’t say I’m any more qualified than you to explain this. But I dipped my toe in and Grothendieck won the Fields medal in 1966.
The Wikipedia article on [algebraic geometry](https://en.wikipedia.org/wiki/Algebraic_geometry) starts off by saying:
>Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
So, it sounds like it started out by solving simple things like 7x+5=0 and then gets a lot more complex.
The Wikipedia on [Grothendieck](https://en.wikipedia.org/wiki/Alexander_Grothendieck) says
><He> was a stateless and then French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called “relative” perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the 20th century.
The reference in Wikipedia led me to [this obit](https://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html), which was interesting, but the math was over my head. But it sounds like he laid the foundation for algebraic geometry and his ideas are still being used today.
I think using Einstein is a way to try and communicate his genius.
Some geometric objects can be described as the zeros of polynomials. For example, you might have heard that the set of points (x,y) that satisfy x^2 + y^2 = r^2 describe a circle with radius r. We can also rewrite that as x^2 + y^2 – r^2 = 0, so the zeros of this polynomial are the points on that circle.
Algebra is the study of mathematical structures like groups and vector spaces. Polynomials are interesting here because they actually form some nice algebraic structures (the set of all polynomials with canonic addition and multiplication actually is a vector space (an algebra even) for example). Examining these structures also tells you things about the solutions of (systems of) polynomials.
The idea of geometric algebra is now to apply discoveries from (abstract) algebra to the representations of geometric objects as algebraic structures via polynomials to solve geometric problems.
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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