What kind of mathematical disciplines would be most useful for physics?

Question

I'm worldbuilding a story, where a famous string theorist hires a student of mathematics to try construct a new theory. For better drama, my premise of the story is that the student never learned more than high school physics.

What kind of mathematical disciplines would be most useful for someone who has never studied relativity nor quantum mechanics to understand it as quickly as possible?

I don't know how important it is, but the basic idea is as follows. A famous physicist visits a small-town college on invitation of an acquaintance who is a professor there. While giving a talk presenting his theory, a latecoming student finds a flaw in the underlying mathematics. The theory is based on a conjecture that is unproven but assumed to be correct. The suggested counterexample gains infamy for the student - something akin to finding counterexamples for the Riemann hypothesis.

Shaken by the latest string of bad news from CERN where experimentalists fail to find supersymmetric particles predicted by his theory, he decides to invite the student to help try a new approach.

I'm not looking for hard science; I'm just looking to tell a good story with some background in math and physics, something like Leonardo making tools in Assassin's Creed.

Answer 1

I wanted to comment on HDE's answer. It's an excellent answer (meaning I agree with it), but I wanted to add the following observations.

First, if you want your character to be a student, you'd best make him a graduate student. As HDE mentions, at the undergraduate level you might have a basic understanding of linear algebra and PDE's, but not enough familiarity to really understand the subtleties.
Second, the fields HDE mentions are related; for example, abstract algebra is (in a sense) an extension and generalization of linear algebra. Differential geometry intersects with it as well. (By the way, to address one of the comments, you can't understand abstract algebra and differential geometry without fully understanding tensors.)
Third, many of mathematical fields mentioned by HDE were motivated by advances in physics, sort of analogous to the way that calculus was invented in part to help solve physics problems. So even though your student may not know much about physics, he would have run across some of the concepts.

Given all that, I think your idea is very reasonable. Many theorists hold supersymmetry to be sacrosanct, but experimentalists aren't quite so sure, and there's always more than one way to describe what we've observed. One historical example I would offer is Feynman's path integral formulation of quantum mechanics, which is a different way of thinking that helped people develop advanced new theories.

(By the way, normally I would just comment, but even though I have the required reputation in Physics, I don't have required 50 reputation here.)

Answer 2

Let's assume that this student wants to begin by understanding the twin pillars of modern physics: quantum mechanics and general relativity. There are several major tools in the toolkit of anyone studying both of these theories at a basic level:

• Calculus (single-variable and multivariable)
  • Differentiation
  • Integration
  • Operators such as divergence, gradient, curl, etc.
• Linear algebra
  • Eigenvalues and eigenvectors
  • Vector spaces, finite-dimensional and infinite-dimensional
  • Tensors
• Differential equations, particularly partial differential equations
• Abstract algebra
  • Group theory
  • Lie algebras and Lie groups
  • Representation theory
• Differential geometry
  • Manifolds
  • Riemannian geometry
  • Metrics and notions of curvature
  • Topology

These are merely starting points that would allow you to understand the basics of the two theories in requisite detail. A physics major might graduate from college with a solid grounding in the first three topics and potentially the basics of group theory. To truly work at the fundamentals of these disciplines may require additional topics such as functional analysis or algebraic geometry.

These tools pop up in most major subfields of physics, not just quantum mechanics and general relativity. They are truly the language of the subject. Most physics majors will walk out of college familiar with calculus, linear algebra and differential equations, regardless of where they go next. Specialization dictates what comes after. If someone goes on to work in relativity, representation theory, for example, may be completely unnecessary. Beyond the list I've given you, it's hard to say what sub-topics would be optimal for this prodigy.

Also important is knowing these tools in context. If you spend years studying group theory in detail, you may be well-prepared to understand the mathematics behind particular parts of quantum mechanics, but you might have little idea what the equations and mathematical structures mean. This is why physics students often learn these tools as they study specific topics. For example, I'm most familiar with the calculus of variations from its use in analytical dynamics, where it is used to derive the Euler-Lagrange equations. In short, it's not just enough to know the math - you must also know the physics.

Answer 3

For basic physics, you need (multivariable) calculus and linear algebra. This is basic literacy. You won't get anywhere in physics without them. There are some differential equations too, but one tends to learn that on a case-by-case basis as one studies examples.

For general relativity, you need Riemannian geometry. Talking about curved spacetime only makes sense if you know what curved spaces are.

For quantum mechanics, you need functional analysis. Everything is phrased in the language of operators acting on Hilbert spaces.

I should warn you that finding a flaw in the underlying mathematics is usually not enough to dissuade a physicist from pursuing a theory. In physics, mathematics is only used to supply a concrete formalism in order to manipulate a concept one is trying to explore.

If the math doesn't quite work out, a physicist will often reason that this means the math being used is not the right mathematics needed to model that theory, and then proceed anyway. They will assume that there is some other mathematics out there (not their job to find out what exactly) which will behave as they like.

Answer 4

For this to work in a good story, you need something that is "bad" mathematics that is used by physicists and easily understood by readers (in its simplest form). The option that comes to mind for me is zeta function regularization, in particular Ramanujan summation. The most famous example of this was "proven" by my favorite mathematician and is below.

$$1+2+3+4+5+6+7+8+...=\frac{-1}{12}$$

This whole topic was developed to define absurd summations like this but is frequently used in the renormalization of quantum field theories. This seems like an area where a physicist might use this willy nilly as a heuristic but a genius mathematician outside the normal circles might have dedicated considerable time to understanding formally.