This question is not very accurate. We can't say which aspects of modern mathematics have been applied to the knowledge of set theory. In fact, most of modern mathematics is based on set theory. Without set theory, there would be no modern mathematics.

However, there are indeed some fields in modern mathematics that are more closely related to set theory, and some fields are not so closely related to set theory. The so-called compactness means that the axioms of set theory play a vital role in some important problems in this field.

General topology is such a field. For example, there is an important "normal Moore space conjecture" in general topology: is every normal Moore space quantifiable? The solution of this problem depends largely on the hypothesis of set theory. Fleissner proved that under the constructive axiom V=L, the locally compact normal Moore space is measurable, and under the assumption of CH or Martin axiom of continuum, an example of unmeasurable normal Moore space can be given. So this conjecture has nothing to do with ZFC. In fact, this conjecture is true under some assumptions of large cardinality.

And the list goes on. In fact, in the fields of topology, algebra, functional analysis, harmonic analysis, combinatorial theory and so on, more and more core problems have been proved to be independent of ZFC system, and to solve these problems, we need to rely on stronger set theory assumptions, such as decisive axiom AD, Martin axiom MA, large cardinal axiom and so on.

However, in the fields of differential equations and differential geometry, people know little about this kind of problems, and most of them can be solved in ZFC. But it is hard to say whether these disciplines will develop in the direction of relying more on the hypothesis of set theory in the future.