I'm not a practitioner, so read with some skepticism, but here's my list:

* Calculus

* Real Analysis

* Statistical Mechanics

* Probability

I'm not sure I have any good recommendations for Calculus, but for real analysis, I would recommend "The Way of Analysis" by Strichartz [0].

I don't have good recommendations for books on statistical mechanics, as I haven't found a book that isn't entrenched in coming from a physics perspective and teaches the underlying methods and algorithms. The best I can recommend is "Complexity and Criticality" by Christensen and Moloney [1], but it's pretty far afield of statistical mechanics and the like. Simulating percolation, the Ising model and ricepiles uses a lot of the same methods as financial simulation (MCMC, etc.).

For probability, I would recommend "Probability and Computing" by Mitzenmacher and Upfal [2], "Probability ..." by Durrett [3] and Feller Vol. 1 and 2 [4] [5] for reference.

I also would recommend "Frequently asked questions in Quantitative Finance" by Wilmott [6].

Also know that there's a quantitative finance SO [7] that might be helpful.

[0] https://www.amazon.com/Analysis-Revised-Jones-Bartlett-Mathe...

[1] https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-Colle...

[2] https://www.amazon.com/Probability-Computing-Randomization-P...

[3] https://www.amazon.com/Probability-Theory-Examples-Durrett-H...

[4] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[5] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[6]