I posted this on r/math, but was told I might have better success here given the level of the question.

Basically, I need to learn how to use the localization theorem to compute integrals on manifolds, and would also heavily prefer to also *understand* the essentials of the theorem and underlying theory. I have read a lot about symplectic geometry and equivariant cohomology the last couple of days, however it has yet to start making sense... I still do not grasp things enough for this theorem/formula to make sense; and at the moment it is ultimately the integration formula I need to make use of, as a computational tool/trick.

Explanations and if possible guided examples of using the formula to compute integrals are very appreciated, and references to any clarifying literature would also be extremely welcome, as the literature I've found hasn't been very friendly when it comes to actually using the technique.

Regarding my knowledge base, I've taken the standard pure math undergraduate curriculum, as well as graduate courses in algebra and analysis, including some topology. My background in geometry/topology is however not as strong as it probably should be for this endeavor. (I do however work towards changing that.)

Thank you in advance