iiinova — where your digital self keeps evolving

Daniel Litt @littmath.bsky.social · 14/09/2026

On what we want to preserve:

In the course of this change, we will have to decide what to hold on to and what to throw away. Some things I would like to preserve: learning seminars; serendipitous conversations that spark an idea; students knocking on a professor’s door to chat about math. A robust community learning exciting new mathematics. Thousands of people that, together, slowly start to resolve their confusion.

I worry that much of what has been written on this topic, including some of my own past writing, focuses too much on trying to preserve the precise shape of the institutions of academic mathematics, rather than our values. How can we preserve the journal and peer review system?5 How can we protect the arXiv? How can we keep our role as gatekeepers? If you have internalized the fact that existing AI systems can produce relatively high quality results for the marginal cost of a few dollars, the idea that any semblance of the current equilibrium can survive what’s coming is absurd.

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Daniel Litt @littmath.bsky.social · 14/09/2026

On pushing buttons and what we can and can't do:

As we think about how to reshape our profession, it’s important to understand that, whether one likes it or not,10 it’s impossible to stop people, amateur or professional, from pushing a button to produce mathematics. The idea that we will persuade people not to play around with math, or that we will be able to “reserve” problems for graduate students, is just not realistic.11 And we shouldn’t want to do this!

There is now more interest in math than at any other time in history. We should be ecstatic for mathematics’s sake, even as we are concerned about mathematicians and mathematical expertise. And by and large, the value of this button-pressing comes from the mathematical community. If a conjecture falls in the woods and no one is around to hear it, who cares?12 For the abundance of new mathematics to have value outside application, we will need an abundance of new mathematicians. And for results with applications, we will want people to be capable of understanding their assumptions and consequences.

I wrote above that solving problems and resolving open conjectures is an incomplete operationalization of our values. But nonetheless it is important to solve problems and resolve conjectures! The provenance of such solutions only matters insofar as it intersects with the existing structure of the profession (incentives, prestige, and so on). It is obvious that structure needs to change in any case.

On balance, I think I like it, though I am sometimes annoyed to find slop PDFs in my inbox. It took me some time to understand that these PDFs expressed a need for understanding; a person elicited them, often without being able to meaningfully engage with their contents, and needed to know that someone could engage, and that someone cared.

That we cannot reserve a problem for a graduate student does not mean we can’t give them the opportunity to work on it. This is compatible with the reconceptualization of a PhD outlined previously.

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Daniel Litt @littmath.bsky.social · 10/09/2026

I think the evidence is very weak. There’s a few plausible ideas floating around; the models can just try them all in parallel. IMO it’s this again:

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Daniel Litt @littmath.bsky.social · 28/08/2026

But the vast majority were real issues, though typically quite minor (typos, ambiguous wording). All of these would be appropriately flagged in a referee report. And I would characterize 5 of them as arguably serious mathematical issues. 8/17

Table of five mathematical corrections, with columns Paper, Location, and Necessary modification.

P13: PDF p. 3, Theorem 1.3, and pp. 23–24, Theorem 7.7(b): replace “degree” with “separable degree.”

P20: PDF pp. 598–599, Theorem 1.10: add the omitted hypothesis p > dim(X).

P20: PDF pp. 602–603 and 625, Theorems 1.20 and 4.29: add the omitted hypothesis that X is projective.

P20: PDF pp. 624–626, Lemma 4.28 and Theorem 4.29: add the omitted hypothesis that D is reduced.

P20: PDF p. 625, Theorem 4.29(3), compared with Theorem 1.20(2): add the omitted hypothesis that Y is proper in the no-rational-curves case.

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Daniel Litt @littmath.bsky.social · 29/05/2026

Just added a 15th problem to problemsilike.com! This one is a "non-abelian" analogue of the variational Hodge or Tate conjectures.

Let 𝑓 :𝑋 →𝑆 be a smooth proper morphism of smooth, connected complex algebraic varieties, and fix 𝑠 ∈𝑆. Let 𝕍 be a complex local system on 𝑋, and suppose that 𝕍|𝑋𝑠 is of geometric origin. Is it true that the restriction of 𝕍 to every fiber of 𝑓 is of geometric origin?

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Daniel Litt @littmath.bsky.social · 12/05/2026

And we have our first proposed solution, to problem #6! I think it is quite likely to be correct, though I am still checking details. I would characterize the solution (produced by GPT 5.5 Pro, prompted by @thomasfbloom.bsky.social as "literature search plus epsilon"). See my preregistered comments:

Let 𝑋 be a smooth projective variety over a perfect field 𝑘 of characteristic 𝑝 >0. Let ℰ be an ample vector bundle on 𝑋 and 𝒢 a coherent sheaf on 𝑋. Let 𝐹 :𝑋 →𝑋 be the absolute Frobenius morphism. Is it necessarily the case that for all 𝑖 ≥rk⁡(ℰ), we have
𝐻𝑖⁡(𝑋,(𝐹𝑛)∗⁢ℰ⊗𝒢)=0
for 𝑛 sufficiently large?

General remarks

Following the terminology of [A04b], this question asks for a computation of the "Frobenius amplitude" of an ample vector bundle in positive characteristic. The notion is motivated by the algebraic proof of Kodaira vanishing in [DI87], due to Raynaud.

The main result of [A04b] shows that, given 𝑋 in characteristic zero, ℰ be an ample vector bundle on 𝑋 and 𝒢 a coherent sheaf on 𝑋, there exists a spreading out of the data (𝑋,ℰ,𝒢) such that modulo almost all primes, the vanishing in the problem statement holds. In positive characteristic, the desired vanishing is only known under strong liftability assumptions [L19], namely that rk⁡(ℰ) <char⁡(𝑘), and that 𝑋 admits a lift
˜
𝑋
to 𝑊2⁡(𝑘) such that ℰ(𝑝𝑁) lifts to
˜
𝑋
for some 𝑁 >0. While this yields some pleasant applications the hypotheses are likely not optimal.

Formalizability

I am not sure if this statement can be formalized given the current state of MathLib.

Some speculation on the problem's difficulty

This is certainly an attention-bottlenecked problem, and I would not be surprised if an answer was already implicit in the literature (or accessible to a frontier model). My expectation is that the answer to the question as asked is "no," though it would be nice to know optimal hypotheses under which the answer is positive.

Comments on interest

The question mostly comes from idle curiosity; as far as I know it has no important consequences.

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Daniel Litt @littmath.bsky.social · 12/05/2026

New project: problemsilike.com, a website collecting open problems that I, personally, like, with comments on their context, difficulty, and interest.

Conjecture 1.1.1 of "Algebraicity and integrality of solutions to differential equations."

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Daniel Litt @littmath.bsky.social · 24/04/2026

I'll finish with a little mystery. The starting point for this work was the classical connection between E_6 and cubic surfaces. There are also surfaces famously connected to E_7 and E_8: del Pezzo surfaces of degree 2 and 1. 14/n

A del Pezzo surface (I think of degree 2?)

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Daniel Litt @littmath.bsky.social · 24/04/2026

Namely, Serre asked if there exist motives with exceptional Galois group; he qualified this question as "hasardeuse." (Serre asked specifically about G_2 and E_8, but later authors have generally understood the question to be about exceptional groups generally.) 10/n

Serre's question: do there exist motives with exceptional Galois group?

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Daniel Litt @littmath.bsky.social · 24/04/2026

The lines on a cubic threefold are parametrized by a (complex) surface, whose real points are pictured below in the case where the cubic threefold is cut out by the equation x_0^3+x_1^3+x_2^3+x_3^3+x_4^3=0. You can play with it yourself here: chocolitt.github.io/fermat_fano_... 6/n

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Daniel Litt @littmath.bsky.social · 24/04/2026

It has long been known that there is some connection between cubic *surfaces* and E_6: loosely speaking, the combinatorics of the 27 lines on a cubic surface are controlled by the Dynkin diagram for E_6. 4/n

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Daniel Litt @littmath.bsky.social · 24/04/2026

One of the crown jewels of 19th century mathematics is the classification (by Killing and Cartan) of compact Lie groups--groups that are also compact manifolds. Dynkin later interpreted this classification in terms of the finite graphs below. 2/n

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Daniel Litt @littmath.bsky.social · 24/04/2026

New paper just dropped, joint with Thomas Krämer and Marco Maculan. It's about a (somewhat mysterious, to me) connection between cubic threefolds and the exceptional Lie group E_6. 1/n

E6-local systems from cubic threefolds
Thomas Krämer, Daniel Litt, Marco Maculan
We produce infinitely many local systems on (level covers of) the moduli space of smooth cubic threefolds, with algebraic monodromy group equal to the exceptional group E6. These local systems arise in the middle cohomology of abelian étale covers of the Fano scheme parametrizing lines in the universal cubic threefold.

Real points of the Fano surface of lines on the Fermat cubic.

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Daniel Litt @littmath.bsky.social · 22/02/2026

Figured I’d re-up this with an excerpt.

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Daniel Litt @littmath.bsky.social · 16/02/2026

Toddler on art during recent museum visits: Left—“They started drawing a house! *matter-of-factly* They’ll finish it tomorrow.” Right—“That baby’s in a tissue box!”

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Daniel Litt @littmath.bsky.social · 05/01/2026

Here's the 4-month moving average:

4-month moving average of new MO questions, steep decline in 2025

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Daniel Litt @littmath.bsky.social · 05/01/2026

New MathOverflow questions each month since the site's beginning. Decline seems to start in ~2021 (due to site moderation changes?), with a notably steeper decline in 2025, since the advent of reasoning models.

New MathOverflow questions each month since the site's beginning; soft decline starting in 2021 and then steep decline in 2025

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Daniel Litt @littmath.bsky.social · 21/11/2025

new trolley problem just dropped: Elon vs. 10^58 randomly chosen people. let's see what Grok has to say

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Daniel Litt @littmath.bsky.social · 24/08/2025

back on the good stuff (geodesic dome tourism)

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Daniel Litt @littmath.bsky.social · 24/07/2025

planning to use this picture in a talk to explain what a foliation is, feeling extremely Canadian

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Daniel Litt @littmath.bsky.social · 19/07/2025

Getting a lot of heat for this on the other site.

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Daniel Litt @littmath.bsky.social · 18/07/2025

it was a perfect summer day in Toronto. the sun was out; the subway station garbage bags danced upside-down in the wind

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Daniel Litt @littmath.bsky.social · 09/07/2025

on the other hand, guitars are predominantly anti-fascist machines

Woody Guthrie with his "This machine kills fascists" guitar.

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Daniel Litt @littmath.bsky.social · 07/07/2025

“we have an ellipsoid at home” / the ellipsoid at home

“ALMOST” stone found in nature, almost an ellipsoid

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Daniel Litt @littmath.bsky.social · 02/07/2025

You can read the paper here: arxiv.org/abs/2507.00167 It even has some pictures.

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Daniel Litt @littmath.bsky.social · 02/07/2025

New paper with my postdoc Simone Coccia!

Density of integral points in the Betti moduli of quasi-projective varieties
Simone Coccia, Daniel Litt
Let Y be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing SL2-local systems on Y with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group.

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Daniel Litt @littmath.bsky.social · 24/06/2025

yes I’m worried about x-risk, why do you ask?

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Daniel Litt @littmath.bsky.social · 23/06/2025

I asked a MathOverflow question. Please tell me if you know the answer! mathoverflow.net/questions/49...

Let 𝐸
be a non-CM elliptic curve over ℚ
. Elkies famously showed there are infinitely many primes 𝑝
at which 𝐸
has supersingular reduction. One may reinterpret this as follows: there are infinitely many primes 𝑝
modulo which the invariant differential 𝑑𝑥𝑦
is locally exact. I am curious about the following:

Let 𝑎,𝑏∈ℚ
be rational numbers. Are there infinitely many primes 𝑝
modulo which the differential
𝜔=𝑎𝑑𝑥𝑦+𝑏𝑥𝑑𝑥𝑦
is locally exact?

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Daniel Litt @littmath.bsky.social · 22/05/2025

Incredibly bad (changes to N.S.F. funding for math):

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Daniel Litt @littmath.bsky.social · 24/01/2025

For example, we are able to check it for many solutions to the Schlesinger Painlevé VI equations (below). 19/n

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Daniel Litt @littmath.bsky.social · 24/01/2025

The main point of the paper is that we can actually prove this conjecture in lots of cases, both for linear, and non-linear differential equations. For example, it's true for the differential equation satisfied by the function F(t) below (which we saw earlier). 16/n

The same globally bounded hypergeometric function from before. It's an elliptic integral!

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Daniel Litt @littmath.bsky.social · 24/01/2025

Just a quick note--it's really important that one takes the Taylor expansion at a point where the differential equation is non-singular. Otherwise there are lots of counterexamples, for example, the one below. 14/n

A globally bounded hypergeometric function

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Daniel Litt @littmath.bsky.social · 24/01/2025

Eisenstein showed that the Taylor expansions of algebraic functions have a special property: only finitely many primes appear in the denominators of their coefficients. We conjecture that this property characterizes algebraic solutions to algebraic differential equations. 12/n

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Daniel Litt @littmath.bsky.social · 24/01/2025

Here's the conjecture. It's not hard to see that a differential equation like the one below (with g a rational function, say) always has local solutions where g is defined--you can just write out their Taylor expansions and check convergence. 11/n

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Daniel Litt @littmath.bsky.social · 24/01/2025

The standard way to formalize it, due I think to Lazarus Fuchs in 1875, is to ask when a differential equation has an *algebraic* solution, i.e. one that also satisfies a polynomial. That is, a function is algebraic if it can be defined implicitly by polynomials. 6/n

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Daniel Litt @littmath.bsky.social · 24/01/2025

The basic question this paper is about is: when can you write down the solution to a differential equation (say, of the form below) explicitly. In fact this has been one of the animating questions of mathematics since the mid-1800s. 5/n

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Daniel Litt @littmath.bsky.social · 24/01/2025

For example, when can you solve a polynomial by radicals? Can you construct a regular n-gon using only a compass and straightedge? 4/n

17-gon constructed by compass and straightedge

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Daniel Litt @littmath.bsky.social · 24/01/2025

New album just dropped: arxiv.org/abs/2501.13175

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Daniel Litt @littmath.bsky.social · 13/01/2025

OK, I think I found the full article -- thank you so much for your help! In case you are curious, ChatGPT gives the following (probably somewhat unreliable) translation of the full article:

A Forgotten Grave?
If one enters the Jerusalem Cemetery from Blücherplatz, passes through the main entrance all the way to the end, then goes about thirty paces to the right around a cluster of family burial sites, and then another twenty paces to the right, one arrives—so a friend of our paper informs us—at a decaying grave. The iron plaque, once set into a stone and bearing the inscription, had long since fallen off and was lying behind the grave. Acquaintances of the deceased’s parents have, with great effort, raised the stone and plaque back up onto the grave. This plaque bears the following inscription:

Ferd. Gotthold Maxim. Eisenstein,
Doctor of Philosophy, Lecturer at the University, Member of the Academy of Sciences in Berlin, Göttingen, and other learned societies.
Born on April 16, 1823,
died on October 11, 1852.

“He lives on for his parents and for science.”

When one sees the condition of this grave, those last words seem like sheer mockery. Yet who was the deceased—still under thirty years of age? We learn this from the following article in the Spenersche Zeitung of October 13, 1852, which reads:

“The Academy of Sciences lost its youngest member, Dr. Eisenstein, day before yesterday through his death. He had reached only thirty years of age, yet he was one of the leading mathematicians not only of Germany but of the entire civilized world!!! After Jacoby’s death, he was probably the only scholar who treated that specialized field of science—number theory—in its full scope, in the broadest…”

“Eisenstein had belonged to the Academy since his youth; he was already recognized as a mathematician of rare talent while still attending the upper classes of a local high school. After a year and a half at university, on Humboldt’s initiative the Philosophical Faculty of the University of Breslau awarded him its doctoral degree. The deceased resembled his previously departed academic colleague Jacoby, of Jewish heritage, although Jacoby had died at a more advanced age. A long-standing hernia ailment, which for years had taken various pathological forms and signaled that the end was near, has now carried off this scholar, who had scarcely reached manhood!”

So much for the article. I would add that, at his funeral, not only all his academic teachers but also Humboldt were present; Gauss likewise came from Göttingen to Berlin for the occasion. His elderly parents looked after the grave for about fifteen years, until they too passed away, and since then it has been completely neglected.

Would it not indeed be an act of proper piety if the Academy were to restore this grave in a dignified manner and see to its upkeep going forward? The costs for such endeavors are rarely all that great. We hope that our appeal may fall on fertile ground—especially since many people are still alive who valued and esteemed this scholar, taken from life so prematurely.

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Daniel Litt @littmath.bsky.social · 12/01/2025

According to German Wikipedia, he was buried at the cemetery at Blücherblatz, but the grave no longer exists. A friend and collaborator in Berlin has visited the cemetery and was indeed unable to find the grave.

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Daniel Litt @littmath.bsky.social · 12/01/2025

I am hoping some of my followers/friends in or near Berlin can help with a bit of historical research. I'm interested in finding the grave of the mathematician Gotthold Eisenstein, or at least information on it.

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