Here is an application to (functional) analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was **whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$**. 

To the best of my knowledge, a first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities in algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic,  see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf
It has far reaching extensions.
The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

The method developed by Bernstein was purely algebraic: out of $|f|^\lambda$ he constructed a module over the ring of differential operators about which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.