**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 $\mathrm{GL}\left(V\right)$ 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 $O\left(N^{N^{N^{N}}}\right)$ of time)

**Finite group classification:** what works for Lie algebras will surely be 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 $\mathbb R\left[\varepsilon\right]$ 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?