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Alex Kontorovich

@AlexKontorovich

Sep 28, 2025

“The Shape of Math To Come”

(with apologies to Ornette Coleman for the title…)

This thread contains slides from (a version of) my talk at last week’s CMSA conference at Harvard, discussing a vision for what research mathematics may look like in the age we now seem to be entering, of AI and formalization.

We will decompose our discussion into three broad categories: (1) the Discovery of new mathematics, (2) the Teaching of mathematics, and (3) the better Understanding of existing mathematics and Communication of it to others. The difference between Teaching and Communication is that the former is meant to apply to newcomers to our subject, whereas by the latter, I mean communication among professionals, perhaps even experts in the same field. I originally wanted to split Understanding and Communication into their own categories but could not find anything that meaningfully distinguished them: any improvement in understanding led to a corresponding improvement in communication, and vice versa.

The old adage “it’s hard to make predictions, especially about the future” (variously attributed to Yogi Berra, Niels Bohr, and others) is very apt here. So we begin by taking a step back.

At the turn of the millennium, some of the world’s most prominent mathematicians gathered in Tel Aviv for a conference to discuss what mathematics might look like in the 21st century. The proceedings were published as a special issue in GAFA called “Visions of Mathematics”, and I strongly recommend one read the whole issue. A particularly prescient contribution was given by Tim Gowers @wtgowers, titled “Rough Structure and Classification”.

Section 2 of Gowers’s paper is called: “Will Mathematics Exist in 2099?” In it, Gowers imagines a “dialogue between a mathematician and a computer in two to three decades’ time” — and here we are! His imagined dialogue is eerily similar to the experiences we all now have on a regular basis:

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Alex Kontorovich

@AlexKontorovich

Sep 19, 2025

Congratulations to Jesse, Jared @jdlichtman, and Christian @ChrSzegedy on this great result! (They told me and Terry about it weeks ago, but released it while I was giving a lecture series in Italy last week, followed by speaking at a conference this week at Harvard -- where I got to chat some more with Jared; so I’m only now getting around to perusing the blueprint+code.) What I’m impressed by:

(cont'd)

• Generating 25K lines of Lean code that compiles! If I understand correctly, *none* of the Lean was touched by humans; the statements as well as the proofs are generated by Gauss. Anyone who’s tried to get 10 or 100 lines of AI-generated code in one shot, knows how many orders of magnitude more difficult that is!
The code is, in some places, highly nontrivial, and even looks nice! Compare their formalization of Borel-Caratheodory to a some places in PNT+ where Thomas Hubert and I collaborated with AlphaProof (which trained via massive RL, and so learned extremely … unorthodox strategies); the latter is completely incomprehensible.

• While there are many, many very short little lemmas, there are also really long proofs; the longest among those I glanced at (namely, `Zeta1_Zeta_Expansion`) is ~150 lines. Wow!

The main new mathematical ingredients, which are of independent interest and should be golfed and upstreamed to Mathlib, are the aforementioned Borel-Caratheodory theorem (a kind of “local” Hadamard factorization), from which one derives a log-derivative expansion (still for general holomorphic functions), and then applies that to the Riemann zeta function. Very, very impressive that this could be done in an automated way. (Basically autoformalized a chapter of Titchmarsh.)

Now some caveats. (Granted: I’ve never tried to raise massive $$ from investors, so can’t say I’d necessarily advertise anything differently in their shoes…)

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Alex Kontorovich

@AlexKontorovich

Jun 16, 2023

then they *would* solve professional-level mathematical problems. (This is not unrelated to issues of "alignment", which I don't have time or expertise to go into here.)

And even if I’m completely wrong, and LLMs are indeed capable of producing, in natural language, something that reads like a perfect math paper, how can we ever trust it? We now have to go and referee these thousands or millions of papers that it can produce and determine which,

if any, are actually correct (as opposed to “hallucinations” - which we mathematicians often refer to as “BS”. Or perhaps "undergraduates" - of which I was one not so long ago!).

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Alex Kontorovich

@AlexKontorovich

Jun 16, 2023

Notes on my lecture at the National Academies workshop on: AI to Assist Mathematical Reasoning. 🧵

I find it useful to work backwards from an end goal; what should be our “Holy Grail” here?

Perhaps it’s that AI should solve the Riemann hypothesis. I see two ways this could go:

1. AI might give a million-line, dense, incomprehensible proof of RH, and I spend the rest of my life just trying to understand what it's saying and why. (Nightmare)

Or:

2. AI might give a perfectly comprehensible, beautiful proof of RH! (Is this a dream? Or also a nightmare?!? Now I'm *really* out of business, and spending my life prompting GPT instead of relishing the thought of solving a super hard problem.)

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Alex Kontorovich

@AlexKontorovich

Nov 14, 2021

@AMathRes

Why I'm excited about @AMathRes: It's as a startup. It's an experiment. Whether it succeeds or fails will be a function of how good its ideas are, and how hard the people that get involved work. (That's why it's rather disheartening to see a coordinated campaign to ...

2/ intimidate founding members into resigning, based purely on ad hominem attacks...) The AMR, again, is only in an organizational phase; the membership (now being assembled) will decide what projects they wish to put their energies towards. That said, here are some of the ...

3/ things I (as a private citizen, not as a Board member) hope we'll end up working on. For background, there have been a number of things we all grumble about but nobody really does anything; for example, democratizing access the world over to high level research mathematics...

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Alex Kontorovich

@AlexKontorovich

Jun 30, 2021

Ugh. Another media frenzy over another purported proof of RH. Usually I just delete such media requests without reply. For some reason (perhaps the fanfare with which this story is spreading), I felt it was my turn to have a quick look and debunk things. TL;DR No, RH isn't proved

2/ If you want to read the paper yourself (which all the press releases seem to not want to let you do; they want you to read the reports of their panels of "experts"- many of whom, if you actually look at the reports, say it's not a proof!...), it's here: researchgate.net/publication/32…

3/ The proof is quintessential in how these go. There is a significant amount of detailed computation on things we teach all our first year grad students, then at some point miracles start happening and details disappear.

The author wants to get cancellation in the Liouville...

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