19th century is watershed in math

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solving equations and symmetry:
polynomials -> galois theory and discrete symmetry
PDEs -> lie theory and continuous symmetry
[and topological spaces / varieties / manifolds generally;
 alg geo is intersection]

i.e., equations are explicitly solvable when there's a nice symmetry;
hence both: understand these symmetries,
and understand the general cases non-explicitly
(structured and unstructured math)

[these arose very practically from trying to solve equations,
 but can also introduce them elementarily]

...and discovered new symmetries
