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If I wanted to do research related to the topology and geometry of space or their applications in quantum mechanics or general relativity, would it be possible to do that from a Mathematician’s POV without the upper level undergad physics typically required?

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Check the website of the program you're interested in. Here's an example.

Basic preparation should include courses in advanced calculus, linear algebra, modern algebra, complex variables, classical mechanics, electromagnetism, quantum mechanics, modern physics, thermodynamics, and statistical mechanics. Knowledge of the following fields is desirable: real analysis, differential equations, probability, topology, differential geometry, and functional analysis.

This makes it sound like you should be OK, but my judgement is not useful; you need to convince the admissions committee.

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For

topology and geometry of space or their applications in quantum mechanics or general relativity

one needs to have a good understanding of tensor algebra, typically used in General Relativity (GR) courses. So if you want to study different topologies of space-time and its implications on GR, then you need to be able to perform matrix and tensor algebra elegantly :)

Research on this can be done from two perspectives - (a) mathematically, identifying the space-time behaviour e.g. conformal field theory, and (b) with a somewhat more physical approach, but that can be in so many different ways that it is peculiar to each project. There is always a non-zero overlap between these two approaches in this research area.

Also, since each programme is unique, they usually will mention the specific courses/skills required to have been taken before they accept the candidates into the project. If that is not mentioned, asking it in the PhD interview is recommended.

Hope this helps.

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  • I'm not trying to be pedantic, but genuinely curious: when you say that there is a "non-zero overlap", is it merely a pleonasm (because overlap itself should indicate that it's non-zero) or is there some obscure field of mathematics/physics that has this weird cases that trained your mind to be specifically say when there is a non-zero overlap, in order to differentiate from those weird cases?
    – cinico
    Commented Oct 1, 2021 at 7:02
  • That's a nice comment @cinico, I think what I want to highlight here is the significant overlap between mathematics and physics, often hard to differentiate which is which and depends on perspective, compared to some other fields of physics where there is substantial use of mathematics but can't be classified as pure mathematics research.
    – Wandering_Alice
    Commented Oct 1, 2021 at 7:09

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