Are new mathematics required to solve problems on the frontier of contemporary physics?

Question

I would like to ask about the following question, and I hope that is suitable. We know that there are relationships between mathematics and mathematical physics, let's say that in physics are used mathematics in the theories of professors that are studying the cosmos.

Currently there are unsolved problems in physics, great problems, for example Wikipedia has the article List of unsolved problems in physics where I recognize some of the most popularized of those unsolved problems. I know about the most famous from an informative point of view.

And I could be right if I think that some of these problems are not (to this date) accessible or attackable because there no exist suitable mathematical tools/techniques to solve or to get a good resollution for these problems. Maybe this thought is naive, I'm asking it from the ignorance.

Question. Are there clear clues that this is the real situation? I mean with (the purpose to clarify) this question that maybe there are clues that support the idea that are required new mathematical methods/ideas to attack these problems. Many thanks.

I would like to know if this can be a good post and if this question is answerable. I'm asking this as a soft question, I don't know if it corresponds to some philosophical point of view. If there are useful references from the literature then refer the literature answering my question as a reference request and I try to search and read those ideas about what clues telling us that for solving some of the problems in the frontier of the contemporary physics will be required new mathematics.

If you think that there is not such relationship, and that there are enough mathematical theories to solve many of those unsolved problems in physics, explain it as an answer.

Since there are several important/popularized unsolved problems in physics, I'm asking just an illustration for some of those in your discussion.

Answer 1

Yes, new mathematics are required. Just as Newton had to invent calculus in order to express his physics more elegantly; so, too, do physicists today have to generalize their mathematics in order to account for new experimental and observational results.

The need for a new mathematics such as Lie-admissible algebras for physics is motivated at a high, philosophical level in:

• Santilli, Ruggero Maria. “An Introduction to New Sciences for a New Era.” Clifford Analysis, Clifford Algebras, and Their Applications 6, no. 1 (2017): 1–119.