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Sometimes when I'm studying a new mathematical topic and I haven't fully internalized it yet, I lose interest easily if I can't connect it to something I already know. For example, I've studied (very elementary) galois theory and a large part of my motivation came from knowing that you can somehow use it to show that general quintic equations can't be solved in radicals (even though I don't know how to prove that yet). To me, a theorem's interestingness is equal to $\frac{\text{how surprising it is}}{\text{how difficult it is to state}}$, but sometimes it's more like $\frac{\text{how clever the proof is}}{\text{how difficult it is to state}}$.

This semester I'm taking courses in Topology, Algebraic geometry and measure theory. I'd love to hear some cool theorems which can be stated in terms of things I've already studied (real analysis, abstract algebra, linear algebra) but require some of my upcoming courses to prove. It doesn't necessarily have to be elementary enough for me to learn in a semester, just needs to be something I can fantasize about.

If you have examples which use other topics I'd be very interested to hear them too.

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    $\begingroup$ Some results that might fit the bill are those from algebraic topology: the "hairy ball" theorem, the Borsuk-Ulam theorem, and the Brouwer fixed-point theorem come to mind. $\endgroup$ Aug 2, 2021 at 23:05
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    $\begingroup$ One topological theorem that is surprising in how difficult it is to prove is the Jordan curve theorem. Once you have the right algebraic-topology tools, however, it's a relatively straightforward proof. $\endgroup$ Aug 2, 2021 at 23:09
  • $\begingroup$ This is a fine question but it's puzzling you've accepted an answer so early. $\endgroup$
    – Integrand
    Aug 2, 2021 at 23:52

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Topology. Too many to count! I'm always partial to the "ham sandwich theorem," stating that given any three compact sets in 3-space, there is a plane cutting all three sets in half (volume wise) simultaneously.

Algebraic geometry. Miles' "Undergraduate Algebraic Geometry" is great at motivating this, but here are some nice problems which you can answer with algebraic geometry and state without it. Bezout's theorem is one of my favorites.

Take $p(x,y)$ and $q(x,y)$ two polynomials of degrees $n$ and $m,$ respectively, whose zero sets define algebraic curves in the Euclidean plane. How many intersections do they have (i.e., points $(a,b)$ where $p(a,b) = 0 = q(a,b)$)?

The answer: As long as there isn't some easy reason that the cardinality is infinite (for example, if $p, q$ are the same polynomial, or $p(x,y) = (x^2+y^2-1)(x^3-y)$ and $q(x,y) = (x^2+y^2-1)(x^2+y^2)$ so that both contain the circle $x^2 + y^2=1$), it is finite, and at most $nm.$ Furthermore, it is actually exactly $nm$, if you count correctly. Here, correct means a) count over $\mathbb{C}^2$ instead of $\mathbb{R}^2,$ so that i.e. $y=x^2+1$ and $y=0$ intersect; b) assign multiplicity to intersections, so that $(0,0)$ is a double intersection of $y=x^2$ and $y=0$; and c), trickiest of all, figure out how to count 'projective intersections' whereby for instance two parallel lines look like they have no intersection, but you can add a 'point at infinity' and say they intersect there.

Measure theory. This is a subject for which this question is easy to answer. The goal of classical measure theory is to figure out how to define the volume of a subset of $\mathbb{R}^n,$ which is a problem so basic that many students don't even realize they've not yet seen a formal solution! Here is a less trivial example, albeit probably too advanced to be discussed seriously in a first course.

In the theory of partial differential equations, the best possible theorem is of the form "For PDEs of this type, there exists a unique solution up to some choice of initial data."

Let's focus on how you can try proving existence results. Now, the space of all differentiable functions is a terribly, terribly behaved space. So you want to work with a space like $L^1_{\text{loc}},$ a space of 'functions you can integrate over bounded sets.' This bigger space will have much nicer analytical properties, making it easy to prove solutions exist.

The only problem: If you have a non-differentiable function, how on earth can you coherently interpret the meaning of a PDE? Well, with measure theory, you can define a notion of weak derivatives, generalizing the ordinary derivative, so that a PDE makes sense even for very wacky functions! This is less of a theorem then a philosophy/approach towards doing PDEs, but it's very fruitful.

If you do want a theorem, a (surprisingly) very related classical result is the isoperimetric problem: Of all the sets in $\mathbb{R}^n$ with volume $1,$ which minimizes the surface area of the boundary? Answer: The sphere, of course! Proving it is the hard part!

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  • $\begingroup$ This is exactly what I was looking for, thanks for taking the time! $\endgroup$ Aug 2, 2021 at 23:42

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