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”
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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.
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