What are some present day examples of mathematics that needs to be made rigorous?

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Hundreds of years ago, calculus, while still very useful, lacked rigor. It was later made rigorous and put on a sound basis by Dedekind and others. Another example is intuitive set theory, which was made rigorous by Zermelo. Are there any present day examples of mathematics that lacks a solid foundation, but which could possibly be made rigorous in the future?