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Daniel Litt Profile picture
Daniel Litt
@littmath
Assistant professor (of mathematics) at the University of Toronto. Algebraic geometry, number theory, forever distracted and confused, etc. He/him.
2 subscribers
Jan 1 • 16 tweets • 3 min read
I've recently been talking a bit about how difficult it is to carefully check even well-written mathematics. I want to try to explain something about this by telling the story of some errors in the literature that (in part) led to the two papers below. 1/n Image
Image These papers began with an attempt to prove an open conjecture in surface topology—the Putman-Wieland conjecture. At some point in mid-2021 Aaron and I were pretty convinced we had a proof, and began writing up the details. 2/n
Dec 22, 2024 • 16 tweets • 3 min read
A couple more brief thoughts on o3’s (incredible) performance on FrontierMath. Image First of all, everything I’m saying is based on the 5 publicly shared problems. I don’t know what the rest of the benchmark looks like, but based on these it’s clearly hard.
Mar 17, 2024 • 17 tweets • 4 min read
The correct answer is “Bob.” Congrats to the 10% who got it right — those few brave dreamers. Image
Dec 30, 2023 • 8 tweets • 2 min read
Here are the numbers from 10 to 100 in alphabetical order:

18, 80, 88, 85, 84, 89, 81, 87, 86, 83, 82, 11, 15, 50, 58, 55, 54, 59, 51, 57, 56, 53, 52, 40, 48, 45, 44, 49, 41, 47, 46, 43, 42, 14, 19, 90, 98, 95, 94, 99, 91, 97, 96, 93, 92, 17, 70, 78, 75, 74, 79, 71, 77, 76, 73, 72, 16, 60, 68, 65, 64, 69, 61, 67, 66, 63, 62, 10, 13, 30, 38, 35, 34, 39, 31, 37, 36, 33, 32, 12, 20, 28, 25, 24, 29, 21, 27, 26, 23, 22.
Nov 30, 2023 • 5 tweets • 1 min read
OK, let me explain precisely what this means and why it's true. Let c be a complex number, and let R be the subring of the complex numbers generated by c. The differential operator

d=(d/dz-c/z)

acts on R[z, z^{-1}].

1/n How does it act? Well,

(d/dz-c/z)z^n=nz^{n-1}-cz^{n-1}
=(n-c)z^{n-1}.

2/n
Sep 29, 2023 • 6 tweets • 3 min read
ChatGPT “proves” the cube root of 27 is irrational, then computes it to be 3, then admits it was wrong about its irrationality, and then finally, when asked to find its error, claims it was right all along. Undisputed king of BS.


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Aug 7, 2023 • 22 tweets • 7 min read
A math thread about some of the questions I've been thinking about lately. The goal is to answer some concrete questions about 2x2 matrices. Our starting point is the equation below--but it will take us on a journey through algebraic geometry, classical analysis, and more. 1/n n matrices which multiply to the identity Everything here is joint work with Aaron Landesman and Josh Lam -- if you're an expert, you can read the paper here:

2/narxiv.org/abs/2308.01376
Apr 25, 2023 • 5 tweets • 1 min read
One of my pet peeves is when a mathematician will claim some other area of math overcomplicates things, or is broadly uninteresting—most math (IMO) is pretty cool, and I think it’s good for the soul to approach other areas with a spirit of curiosity, rather than superiority. Of course it’s fun to post a screenshot of some complicated text and tweet “math is the language of the universe lol” or something—I’ve done this myself—but I think it’s best done in with a bit of self-mockery, rather than playing into some imagined rivalries.
Apr 19, 2023 • 22 tweets • 3 min read
In this thread I will try to explain what a short exact sequence is, with an interested 8th grader as the target audience. All I'll assume is some familiarity with systems of equations, which is typically a middle school topic. 1/n Most of you think this is impossible (see the poll below)! But I think it can be done, admittedly with some imprecision.
Oct 13, 2022 • 4 tweets • 1 min read
One of the most misleading things about learning mathematics is that you are seeing the *output* of a great deal of trial, error, sweat, and tears, presented as if it’s obvious, or at best a product of cleverness. I often try to explain how one might have discovered the proof I’m presenting, but even so one can’t show all the blind alleys one would go down in actual research.
Oct 12, 2022 • 4 tweets • 1 min read
Continued studying games in my Intro to Proofs class today, and one of my students beat me in a game! Couldn’t be more proud :). (i’m not owned, i’m not owned…)
Jun 13, 2022 • 42 tweets • 10 min read
A thread on some of the math I've been thinking about lately. I think it's really fun: it involves connections between low-dimensional topology, algebraic geometry, classical analysis and ODEs, dynamics, representation theory, number theory, and more. 1/n This thread is meant for people with some background in math -- maybe partway through a math major, plus or minus a bit. If you're a professional, you can read the intro to the relevant paper here: arxiv.org/abs/2205.15352
2/n
Jun 11, 2022 • 4 tweets • 1 min read
Thinking about this beautiful thread again. If I understand correctly the proof works (essentially) by arguing that certain primes are not square in the 2-adic numbers; this cannot work for primes which are 1 mod 8, as they are 2-adic squares. And 17 is the first such prime! This suggests that in some sense the proof sketched in the thread does generalize to arbitrary primes p; one just has to choose the right auxiliary prime such that p is not a square in Q_l. And there are infinitely many such, by Chebotarev.
Mar 9, 2022 • 8 tweets • 6 min read
@YuriSulyma @sarah_zrf @SC_Griffith I don’t know a good de Rham-Witt formalism; things are much more subtle in positive/mixed charactetistic, and there are multiple inequivalent notions of connection (e.g. flat vector bundles are no longer the same as O-coherent D-modules)… @YuriSulyma @sarah_zrf @SC_Griffith I recommend Chapter 9 in Grothendieck’s “Dix Exposés” for one nice PoV on this. Basically the point is that (in char 0) a connection gives a rule for extending sections along nilpotent thickenings (that is, Taylor expansion), and you can formulate this as a sheaf on the site…
Dec 4, 2021 • 11 tweets • 8 min read
Different images generated by an AI art app (wombo.art) with the prompt “encyclopedia of alien botany” Some more:
Dec 19, 2020 • 4 tweets • 1 min read
i call it "silicon valley" yo i solved the trolley problem, it was easy
Nov 18, 2020 • 26 tweets • 7 min read
A thread on the equilibria of pendulums and their connection to topology. 1/n Consider the following physical system—an n-uple pendulum. This is a bunch of rigid rods connected at ball and socket joints, each of which has a weight, and pinned in place at the top. 2/n
Aug 5, 2020 • 29 tweets • 7 min read
A short thread about how infinitesimally small spaces (called non-reduced schemes) can appear in the real world. We can build them out of ball-and-socket joints and rigid rods. 1/n Here are the basic building blocks. The black and red dots are ball-and-socket joints, and the black line segment is free to spin around them, as long as its length remains the same. But the red dot is pinned to the page, while the black dot is free to move. 2/n
Jul 25, 2020 • 4 tweets • 1 min read
i...uh...i think GPT-3 is tired of me trying to get it to do math this is real. I’m shook.
Jul 14, 2020 • 33 tweets • 9 min read
Thread, lost due to computer crash and recreated: a mysterious connection between number theory (the analytic class number formula) and algebraic topology (the Dold-Thom theorem). 1/n The class number formula (below, from Wikipedia) tells us that the residue of the Dedekind zeta function of a number field K at s=1 is related to certain more algebraic invariants of the number field -- for example, its class number h_K and the number of roots of unity in K. 2/n
Jul 2, 2020 • 13 tweets • 2 min read
A short thread on the philosophy of "pathology" in mathematics. This is the serious version of the tweet below. 1/n As far as I can tell there are two things a mathematician may mean if they're talking about pathology:

(A) if you drop some hypothesis (characteristic zero, noetherian, finitely generated, etc) things become much wilder, and

(B) something you hoped was true but isn't.

2/n