Representation theory of Lie groups: there is a whole world between $mathrm{Sym}^n V$ and $wedge^n V$. (Okay, this is an oversimplication - I am talking about the representations of $mathrm{GL}left(Vright)$ here, but this is the fundament of all other classical groups.)
Constructive logic: if you can't compute it, shut up about it. (At least some forms of constructive logic. Brouwer seemed to have a different opinion iirc.)
Homological algebra: How badly do modules fail to behave like vector spaces?
Gröbner basis theory: polynomials in $n$ variables can be divided with rest (at least if you have some $Oleft(N^{N^{N^{N}}}right)$ of time)
Finite group classification: what works for Lie groups will surely be even simpler for finite groups, right? ;)
Algebraic group theory: In order to differentiate a function on a Lie group, we just have to consider the group over $mathbb Rleft[varepsilonright]$ for an infinitesimal $varepsilon$ ($varepsilon^2=0$).
Semisimple algebras: The representations of a sufficiently nice algebra mirror a structure of the algebra itself, namely how it breaks into smaller algebras.
$n$-category theory: all the obvious isomorphisms, homotopies, congruences you have always been silently sweeping under the rug are coming back to have their revenge.
Modern algebraic geometry (schemes instead of varieties): let's have the beauty of geometry without its perversions.
How many of these did I get totally wrong?
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