What are streams of mathematics that are generally not used by Physicists but you think can be used successfully provided they become more developed? I want to know about such things cause it is quite interesting. Also, can Projective Geo be used in physics?
There is a contradiction within your question. 'Unknown' and 'which methods' cannot be answered simultaneously. Maybe we should change the title and directly ask for projective geometry in physics?!
String theory which is based on graded Lie algebras are an open problem, but it has known methods, whereas the physical significance is missing.
The cosmological investigations of de Sitter and anti-de Sitter spaces can be seen in the context of algebraic topology, but it is again known methods with open questions in physics.
So how should an answer look like? As soon as we can accurately describe a problem in physics, as soon do we have methods at hand, namely those the problem is described by. Whether such a problem can be dealt with new mathematics is per definition unknown, so cannot be answered.
There are some fundamental limitations and it is unclear whether they are necessary or only the usual way. Physics is done in frames, that is we have coordinates and we can measure quantities. These are strong restrictions since many mathematical objects have neither a coordinate system nor a metric, and I'm not aware of a physical question which doesn't expect them.
I remember that I once asked on PF why the Lie groups and Lie algebras in physics are always (I know, Heisenberg and Poincaré are exceptions here) the semisimple ones? Why don't their big solvable subalgebras play a role? Why do we always need to consider operators which walk the ladder up and down? The best answer I received was, that those semisimple cases bring along a metric, something to measure with geometric methods. Now does this mean we are simply used to rely on measurements and do not consider other possibilities, or is it a physical requirement? This is hard to answer, if at all. But it is basically a version of your question.
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