> Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators

Linear combinations of a basis (b1, b2, ..., bn) are sums of the form a1*b1 + a1*b2 + ... an*bn using some coefficients (a1, a2, ..., an). Convex combinations add the requirement that a1 + a2 + ... an = 1. Algebraic operations adds other things besides adding and multiplying by coefficients. Anyway, he's saying that if you have a function for which f(a1*b1 + a2*b2) = a1*f(b1) + a2*f(b2), then you only need to know what f does to b1 and b2 and the other basis elements in order to know what it does to anything.

    > Multiplicative structure (in analytic number theory) -> suffices to check prime powers

Here he's talking about how (for example) some functions f(ab) = f(a)f(b) (sometimes with the extra condition that `a` and `b` have no factors in common).

For such functions with "multiplicative structure", if you want to know their value at any point, you only need to values at prime powers.