I promised some maths so here comes. The overarching theme for these posts is a framework called deformation/rigidity (you have to give it to us, we have pretty cool names) introduced by Sorin Popa in the early 2000's. One crucial aspect here is the notion of an s-malleable deformation. I'm trying to keep things intuitive for now so I won't include yet the actual definition but some intuition below:
Naive definition: One starts with an action of a group G on a von Neumann algebra, say (A, τ). The commutative counterpart would be an action of a group by (potentially discontinuous) Borel automorphisms of [0,1] which preserve the Lebesgue measure - this is in fact a much more general example than one might expect. Then an s-malleable deformation of this action consist of an action of G on a larger algebra (𝔄, τ) which contains two copies of A, sitting 'orthogonal' to each other inside 𝔄 and which can be 'deformed' into one-another by a continuous path of automorphisms that commute with the G-action. Furthermore we ask that this path is symmetric in time.
There will ideally be a number of posts explaining why anyone would care and what this is useful for but what I want to draw attention to is this idea of 'giving yourself space' to work by putting your object inside a larger object. It's really ubiquitous in maths I feel but somehow I get excited every time I run into it even if I've seen it so many times already (maybe because it inevitably goes wrong whenever I try to use it myself).
So really I just though I'd draw attention to this kind of phenomena:
The Lebesgue integral and DCT. At the end of the day, every time I want to compute an integral I do it by reducing to a Riemann integral so clearly the point of the Lebesgue integral is not that it's computable, but that by developing this fancier language you can prove things which are true for the Riemann integral itself but which you couldn't prove before - e.g. without secretly using the Lebesgue measure or Fatou's Lemma try proving that if a sequence of Riemann integrable functions on [0,1] is unformly bounded and converges pointwise to 0 then the integrals converge to 0. You need 0 knowledge about the Lebesgue measure to state it but good luck proving it without the language of Lebesgue (*I am not saying it's impossible but rather non-trivial I think)
Very often in number theory/diophantine equations. Just one example which I've seen recently: say a is a positive integer and consider n=a^3-3a+1. Prove all prime factors of n are either 1 (mod 9) or -1 (mod 9). Clearly you'll want to work in 𝔽_p but really you'll have to move to a degree-2 extension of 𝔽_p in order to write a=t+t^(-1) which will generally not have roots in the base field. (It's a cute exercise to finish the problem for here)
Furstenberg's topological proof for the infinity of primes *but also* his proof of Van der Waerden's Theorem (any way you partition the naturals into finitely many subsets one of them contains arithmetic progressions of arbitrarily long lengths) using the Poincare/Birkhoff multiple recurrence theorem from ergodic theory.
Bernstein's p^αq^β theorem using rep theory but also a bit of Galois theory.
Even the basic Cauchy's Theorem from group theory is just a really cute counting argument, I wasn't aware combinatorics was invited to this party??
I'll leave you with two more examples. They both can be done with func-anal tools but you wouldn't expect it. It's quite fun thinking about them:
Let G be a discrete group acting on a set I. Prove that the following are equivalent: 1. all orbits are infinite and 2. whenever A and B are finite subsets of I there is some g in G such that gA is disjoint from B. As an aside, if you endow I with the counting measure this is saying that the action is ergodic iff it is weakly mixing but this is rather irrelevant for the proof.
(This one you can also do with martingales I think but also there is a functional-analytic proof) Suppose on each lattice point (m,n) of the plane you write a real number a_{m,n}. You know that the bi-infinite family (a_{m,n})_{m,n} is bounded and that every number is the (arithmetic) mean of its four neighbours. Show that in fact all the a_{m,n} must be equal.
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