The correct answer is “Bob.” Congrats to the 10% who got it right — those few brave dreamers.

https://x.com/s1lknoose/status/1769177202816434655?s=46&t=41cpGRZavPniB0OfiIMsDA

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.

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

https://twitter.com/littmath/status/1729996751568859627

How does it act? Well,

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

2/n

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.





o no

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 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

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.

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.

https://twitter.com/littmath/status/1648315697791942658

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.

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 :).

https://twitter.com/littmath/status/1577799621920915456

(i’m not owned, i’m not owned…)

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

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!

https://twitter.com/andresecaicedo1/status/1302697810446483456

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.

@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…

Different images generated by an AI art app (wombo.art) with the prompt “encyclopedia of alien botany” Some more:

i call it "silicon valley" yo i solved the trolley problem, it was easy

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

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.

https://twitter.com/ChocoLinkage/status/1266326082061340672

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

i...uh...i think GPT-3 is tired of me trying to get it to do math this is real. I’m shook.

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

A short thread on the philosophy of "pathology" in mathematics. This is the serious version of the tweet below. 1/n

https://twitter.com/littmath/status/1278116032880824326

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

Today I am turning 32 — probably the last time my age will be a fifth power! In celebration, here’s a thread on one of my favorite techniques in mathematics: “spreading out.” It’s one of the sources of the many mysterious connections between geometry and arithmetic. 1/n Let me start with a simple example, due to Serre. Suppose f: ℂ^n—>ℂ^n is a polynomial map such that f ⚬ f = id. That is, f is it’s own inverse.

Then I claim that f has a fixed point. 2/n

In celebration of reaching 3k followers, here's a thread on the Sylow theorems and algebraic geometry. I'll start by recalling the Sylow theorems, and then explain how they are in some cases manifestations of the geometry of certain algebraic varieties. 1/n Let G be a finite group (say, the set of symmetries of some object) and p a prime. If the size |G| of G is p^k*m, where m is not divisible by p, we call a subgroup of G a *p-Sylow subgroup* if it has size p^k.

The first Sylow theorem says p-Sylow subgroups always exist. 2/n