I asked two questions on twitter yesterday. The previous post summarized the results for a question about books that I asked from my personal Twitter account.

This post will summarize the results of a question I asked from @AnalysisFact.

If a genie offered to give you a thorough understanding of one theorem, what theorem would you choose?

— Analysis Fact (@AnalysisFact) June 15, 2018

Here are the results by category. Some are not strictly theorems but rather topics or conjectures.

## Computer science

- Curry-Howard correspondence
- P=NP or not
- Collatz Conjecture

## Logic

- Cohen’s forcing theorem
- Godel’s Incompleteness theorem
- Continuum hypothesis
- Zorn’s lemma

## Algebra and number theory

- The ABC conjecture (theorem?)
- Prime number theorem.
- Riemann hypothesis
- Fundamental theorem of finite abelian groups
- Classification of finite simple groups
- Fermat’s last theorem, the unpublished Fermat version

## Topology and differential geometry

- Thurston’s geometrization conjecture
- Gauss Bonnet theorem
- de Rham’s theorem
- Grothendieck Riemann Roch theorem

## Analysis

- Banach-Tarski theorem.
- Stokes theorem
- Carleson-Hunt theorem
- The epsilon/delta definition of continuity
- Universal approximation theorem

## Differential equations

- Navier–Stokes existence and smoothness postulate
- The relativistic version of the Shrodinger equation
- Atiyah-Singer index theorem

## Mathematical physics

*E*=*mc*²- Noether’s theorem
- Liouville’s theorem

## Miscellaneous

- Existence of general equilibrium prices
- Farkas’ lemma
- The graph minor theorem
- Central limit theorem

## Mischievous genie

A couple people picked up the fact that in folk stories, being granted a wish doesn’t usually turn out well.

M. Shah: uh oh. Is this one of those malicious genies that twists language used to make the wish so that you are granted some horrific wish?

Jumpy the Hat: You now understand every single thing about irrational numbers but it’s all wasted because you’re cursed to become NJ Wildberger and you don’t think they exist

M Shah: or you want to thoroughly understand some theorem about Weierstrass’s monster. But little did anyone know that Weierstrass actually did have a pet monster. And it ends up biting your head off because it doesn’t like other things that are continuous.

Love this, thank you! Glad ‘my answer’ of Atiyah-Singer turned up from someone else.

A lot of people misunderstood the question and answered conjectures instead of theorems. Riemann’ hypotheses isn’t a theorem—it might not even be true!

The vague ones, like the Reimann hypothesis, don’t bother me. Seems like fair game to want to thoroughly understand a theorem without necessarily knowing exactly what the theorem says. For example, one might wish to thoroughly understand whatever theorem most effectively resolves the P=NP question, without knowing how that would be resolved. (I’ve wondered whether it could be proven that there’s no way to prove P=NP either true or false. Possibly as some sort of outgrowth of incompleteness of the continuum.)