“If a genie offered to give you a thorough understanding of one theorem, what theorem would you choose?” blogger John D. Cook recently asked on his @AnalysisFact Twitter account. Responses ranged from the names of theorems to questions about the genie’s potential trickery to creative ways of gaining insights beyond an understanding of one theorem.

Cook summarized the results on his blog. In an interview conducted over email, he answered my questions about his genie idea. (The following interview has been edited for length and clarity.)

**Rachel Crowell: How did you come up with the question? **

John D. Cook: I asked two similar questions on Twitter around the same time. One was about what book would you like to have read but don’t want to put in the effort to read. The other was what theorem you’d like to know without having to study. So both are related. I don’t remember which I thought of first.

**RC: I saw your own response to the question: “For a long time I would have said the Atiyah-Singer index theorem. Now maybe I’d say the classification of finite simple groups, especially how ‘the monster’ group relates to unexpected things.” Would you please elaborate on that point?**

JDC: As for my earlier interest in the Atiyah-Singer index theorem, I was a grad student at the time, and that theorem seemed to be the culmination of many of the things I’d studied. I thought that if I understood that theorem, I’d put a lot of pieces into place.

My interests have changed since then. I don’t think about differential equations on manifolds much anymore. Now I think more about probability and statistics.

I have no professional interest in the classification of finite simple groups. That’s one reason I find it interesting, because it is so unfamiliar. It’s curious that it’s an easy problem to state, starting from scratch, and yet it has taken a tremendous amount of effort. It’s also curious that the sporadic groups have unexpected connections to sphere packing, physics, etc. You have these groups that were the last to be discovered, the odds and ends that don’t fit into the rest of the theorem, and they turn out to be the most interesting.

The thing I enjoy most about math is seeing surprising connections. That’s what the Atiyah-Singer index theorem and the classification of finite simple groups have in common: they form deep connections. That’s also why I enjoy applied math. I get to see not only connections between areas of math, but also between math and diverse applications.

**RC: ****What else would you like to share about your blog, your social media accounts or your background?**

JDC: If someone would like to find me online, the two best places are by blog (https://www.johndcook.com/blog/) and my Twitter accounts (https://www.johndcook.com/twitter/).

I also posed the genie question over email to two other bloggers: Patrick Honner (@MrHonner) and John Golden (@mathhombre):

“I would ask the genie to grant me full understanding of Cauchy’s Integral Theorem,” Honner said. “I’ve always found complex analysis fascinating, and intimidating. I’d love to better understand complex integrability and differentiability, and Cauchy’s Theorem seems like one of those concepts that unlocks a lot of related mathematical understanding. Especially since it’s connected to ideas–like the Fundamental Theorem of Calculus and vector analysis–that I know something about,” he added.

“For me, I think it would be the Atiyah Singer Index Theorem,” Golden said. “I love results that connect fields of mathematics, and this is the beauty of the genre. The connection to the Euler characteristic, the birth of K-theory, the first proof sketched but then changed before publishing … so many great stories. Plus my advisor instilled a lasting appreciation for Sir MFA’s writing. Though, I think, I migrated from math to teacher education immediately after graduating. With that hat on, I think about … well, too much. The classification of the platonic solids, the quadratic formula, the fundamental theorem of algebra… maybe I’d better stick with my mathematician-y answer. And understanding the undecidability of the Continuum Hypothesis!” he added.

Which theorem would you ask the genie to help you magically, thoroughly understand? Share your responses in the comments below! You can also find me on Twitter @writesRCrowell!

P.S. My answer? I would ask the genie to grant me a thorough understanding of Shinichi Mochizuki’s approximately 550 page proof for the abc conjecture, a topic I’ll discuss in my next post later this month. Even though my request is related to a conjecture and not a theorem, I would hope the genie would still grant my wish.

Personally, I’m working on Collatz Conjecture, so that’d be cool to know. But as long as we are granting wishes, Riemann Hypotheses is the easy answer for me.

At the Workshop on Infinity and Truth at the National University of Singapore in 2012, the organizers asked the conference participants to all pose a question they would ask to an all-knowing mathematics oracle, which is similar to this question. The results were published in the conference proceedings, and my contribution also appears at http://jdh.hamkins.org/question-for-the-math-oracle/.

I would choose the Jordan Curve Theorem, but with the caveat that I want the analysis-type proof not the homology proof. It shows up in your first graduate course in differential geometry and the statement of it is so simple that my three year old can get the ‘basic’ idea. But then it’s surprisingly difficult to prove and requires a fair amount of technical machinery which I have completely forgotten. This doesn’t make any sense! I love it! And then you learn the homology proof and realize that different fields of math are so powerful in different ways.

You mean the “proof” of the abc conjecture by Mochizuki which is now widely suspected to be completely void of content? In that case, your wish may well come true if there really is nothing there to understand!

an exact integral solution of the general homogeneous canonical 2nd order linear ordinary differential equation with arbitrary variable coefficients ?

@Cherry Blossom: “is now widely suspected to be completely void of content” that is something new in the rumors. The last state publicly known is that Scholze and Stix had some concrete objection, which according to Fesenko is regarded from the “other side” as “shallow and misplaced”. “Completey void of content” is new.