Paul Fabel

AI Crushed the Math Olympiad—Or Did It? AI models supposedly did well on International Math Olympiad problems, but how they got their answers reminds us why we still need people doing math

When you (and especially, when _I_) don’t have a clear understanding of something, the best approach is to read what Emily Riehl writes about it. #MathSky

www.scientificamerican.com/article/math...

AI Crushed the Math Olympiad—Or Did It? AI models supposedly did well on International Math Olympiad problems, but how they got their answers reminds us why we still need people doing math

@sciam.bsky.social gave me the opportunity to share some personal thoughts about the recently reported AI results from the #imo2025:

www.scientificamerican.com/article/math...

I can definitely feel it and admitting that is no defense of the (also relatively empty) hellscape that is X.

I wish people would use the repost button more than the like button to boost content they appreciate, which might help. Encouraging activity by news media and journalists also might help.

Every unbounded countable ordinal X has a terminal sequence x1<x2<x3....

But each interval [x_(n),x_(n+1)] is (uniquely) order isomorphic to a PROPER initial subset of X.

Basic but useful in the context of transfinite induction.

Let X denote the colimit indexed over the naturals, of N, NxN, NxNxN,....

Each factor with the dictionary order, and N is omega_0, a minimal infinite well ordered set.

With the order topology, deleting the isolated points of X yields a space uniquely order isomorphic to X.

Deleting the isolated points from omega_1, leaves us with a well ordered space uniquely isomorphic to omega_1.

Here, omega_1 is a minimal uncountable well ordered space.

But can this happen for a COUNTABLE well ordered space?

YES!

Patterns in math that work until they don't:

Start with a well ordered topological space with the order topology.

Now, delete the isolated points. Creating a closed subspace.

Repeat.

Did you get an empty space after finitely many iterations?

Works every time until it doesn't.

1984 was about a half hour ago, Ryne Sandberg emerged as the best player on the planet.

What a Week! July 26, 2025 "SARAH KENDZIOR" Edition! You can’t be well informed unless you are informed well! Join hosts Erik Nelson and Tony Russomanno with guest Sarah Kendzior as they unpack the week that was, on this July 26, 2025 “SARAH KENDZIOR!” ...

New interview! On Trump, Epstein, and much more: ksqd.org/what-a-week-...

Sweet Hereafter (1997) >> Boogie Nights (1997).

Leaving Las Vegas (1995) >> (Heat or Toy Story)

do you ever stare at the ceiling and think about how the worldwide scientific establishment did the impossible and created a COVID vaccine in under a year and the response of the general public has been to go on an unstoppable rampage to destroy science and scientists

YouTube video by Sheafification of G What's the big deal with the Yoneda Lemma?

Yoneda Lemma?

Sheafification G.

5 stars. Highly recommended.

www.youtube.com/watch?v=q23H...

Waiting Around to Die

Closing music.

1) Breaking Bad. (S2E3 Bit by a Dead Bee)
2) The Be Good Tanyas (performers)
3) Townes Van Zandt (composer)

Am I teaching general topology Fall 2025?

I DO beg your pardon.

I am teaching topology and its APPLICATIONS

to computational topological data structures.

Topics may include classification of Cantor Space,
an exploration of the computational reals,
and MAYBE a sojourn into Erdos space.

Am I teaching number theory Fall 2025?

I DO beg your pardon.

I am teaching number theory and its APPLICATIONS

in modern computational cryptography.

A totally disconnected metric space X might NOT have a basis of clopen sets.

However if X is ALMOST 0-dimensional, then X embeds in some R-tree. (Oversteegen and Tymchatyn)

www.ams.org/journals/pro...

Erdos space is the subspace of all rational points in l_2

Since projection maps are continuous, Erdos space has LOTS of clopen subsets.

However every clopen nbhd of 0 is unbounded.

This is a little surprising since E is not locally compact, and boundedness is not a topological property.

The Trump gang took down a bunch of legally mandated climate assessments. Meanwhile, in actual reality, the climate doesn't give a fuck:

“We are witnessing a true reversal of ocean circulation in the Southern Hemisphere—something we’ve never seen before. While the world is debating the […]

So Shines A Good Deed In A Weary World Dark isn't the opposite of light, it's the absence of light.

On my country's 249th birthday, I wrote about the problem of sharing a nation with millions of evil people, and how to live a human life in the face of overwhelming inhumanity.

Oh and the Beastie Boys make an appearance.

www.the-reframe.com/so-shines-a-...

Erdos (1940) exhibits a totally disconnected metrizable separable topological group which does NOT have a basis of clopen sets.

The proof is straightforward, and accessible to an advanced undergraduate.

www.renyi.hu/~p_erdos/194...

Counting with Categories by John Baez:
Part 1: golem.ph.utexas.edu/category/202...
Part 2: golem.ph.utexas.edu/category/202...
Part 3: golem.ph.utexas.edu/category/202... 🧮

TFAE if p>2 is prime.

1) All solutions to x^8=1 mod p are a power of 2.

2) p=8k+5

3) x^8=1 mod p has exactly 4 solutions mod p.

Thankyou Quadradic Reciprocity Supplement.

Identity Sprain Trump’s birthday parade In her masterpiece Secondhand Time: The Last of the Soviets, the Nobel-Prize winning author and oral historian Svetlana Alexievich...

(a) @swordsjew.bsky.social knocks it out of the park with this one.
(b) I should read more Alexievich. Voices from Chernobyl was one of the most affecting, well-crafted books I’ve encountered.

buttondown.com/theswordandt...

NO KINGS STARKVILLE · No Kings **In America, we don’t put up with would-be kings.** NO KINGS is a national day of action and mass mobilization in response to increasing authoritarian excesses and corruption from Trump and his allie...

NO KINGS in Starkville, MS, Saturday June 14. 12-1:30 pm
N. Jackson and E. Main.

www.mobilize.us/nokings/even...

We get a 2 parameter ring R of 2x2 matrices generated the real diagonal matrices, and the matrix d with d(x,y)=(y,0).

We have d^2=0 and thus the inverse d^(-1) does not exist.

Can we try to do synthetic differential calculus in R anyway?

For more context, terrific notes by Mike Shulman, hyper-reals in the context of synthetic differntial geometry.

home.sandiego.edu/~shulman/pap...

Struggling to land the trick or idea that will make your proof work?

Just make everything MORE complicated!

That way, nobody will EVER understand it.

Infinitesimals happen on the road to constructing the additive reals.

Start with the free abelain F group over Cantor Space.

Mod out by the relation g= 2s(g) where s:F-->F is rightward shift. Call the quotient G.

Let d=1000... - (01111...) in G. Then d^2=0.

G is isomorphic to R+dR

Since F is abelian, f(x)=2s(x)-x is a homomorphism.

Mod out by the image of f, and we get...a model of the additive hyper-reals?

We must mod out by a bit more to deduce that 1000000.... is equivalent to 0111111....

Given an endmorphism f:A-->A of an abelian group,

f(A) is normal in A, and hence A/f(A) is a group.

This easy fact helps speed run a construction of the additive reals as quotient of the free abelian group F over Cantor space.

Identify x with 2s(x) with s:F-->F rightward shift.