If X is a separable Banach space, then the unit ball of X* is metrizable in the weak* topology! This fact plays a significant role in the theory of Banach spaces, iirc.

As a concept, it comes up pretty often in basic functional analysis. Urysohn's theorem in particular is probably not so important, but it is very cool imo.

The fact that there are so many probability paradoxes should make it clear that probability was a mistake. Things either happen or they don't, and we'll just have to wait to find out which. Be ye not tempted by sorcery.

I've arrived in DC for JMM!

DM me if you're around, I'd love to grab coffee etc :)

Yes I love multisets

Haha I should've looked at your full name

I think where you should start depends a lot on how much background you have with commutative algebra (and comfortability with rings / modules in general) -- how do you feel about these things

Happy new year, all :)

Hardy came to visit Ramanujan in the hospital on New Year's Day.

"On my way here, I noticed that the current year is 2026. A very uninteresting number."

"On the contrary! 2026 is the 40th smallest positive integer which is expressable as the sum of 7 cubes in at least 9 ways."

Literally true, check out my papers 😎

bsky.app/profile/moti...

The Formal Context of Saturated Transfer Systems on Finite Abelian Groups We describe the reduced formal context of the lattice of saturated transfer systems on a finite abelian group. As an application, we compute that there are 13,784,538,270,571 saturated transfer system...

I worked with UVA undergraduate Seth Bernstein on this fun homotopical combinatorics project: arxiv.org/abs/2511.02982

He'll be presenting a poster on it at this year's JMM! meetings.ams.org/math/jmm2026...

So, γ' is also a unit-speed parametrization of the unit circle! In particular, we have γ'(t) = γ(t+π/2), i.e.

cos'(t) = cos(t+π/2) = -sin(t)

sin'(t) = sin(t+π/2) = cos(t)

Now consider the parametrization γ of the unit circle defined by

γ(t) = (cos(t), sin(t)).

This parametrization has constant speed 1 (by definition, if you'd like!)

That means γ'(t) is a unit vector for all t, and we know it is orthogonal to γ(t) for all t by the GEOMETRY FACT.

A yellow circle with center O, and tangent line T to the circle. The line segment (radius) from O to the point of intersection between T and the circle is shown. The radius is orthogonal to T.

Same as the e^{ix} thing but said differently:

We start with a 💥GEOMETRY FACT💥

A tangent line to a circle at a point p is orthogonal to the radius of the circle at p.

Idk but looks kinda weird. I found this PDF, which seems to be from the same "Wallot": www.leonschools.net/cms/lib/FL01...

On the ωth day of Christmas my true love gave to me

ω numbers natural

...

And a partridge in a pear treeeeee

GitHub - Macaulay2/M2 at development The primary source code repository for Macaulay2, a system for computing in commutative algebra, algebraic geometry and related fields. - GitHub - Macaulay2/M2 at development

The package will be included in the next Macaulay2 release (scheduled for November I think). Or you can grab it from the development branch to install it now!

github.com/Macaulay2/M2...

I think this will genuinely save equivariant homotopy theorists a lot of time and hair-wringing, I'm so stoked.

Very very happy with this project we ran at the M2 workshop this summer in Madison -- it is now possible to do compute Ext, Tor, etc. of C_p-Mackey functors by computer!

The image below shows how you can use the package to compute a free resolution of a C_p-Mackey functor.

"What can we do about this? Simply choose to live in the worst of both worlds."

I'm interested (for weird reasons) in the asymptotics of this expression as n,m → ∞

And more generally in the distribution of the number of such pairs (A,B), but that seems much harder than just studying the mean.

What is the expected number of pairs (A,B) with A⊆{1,...,n} and B⊆{1,...,m} such that

(i) X_{i,j} = 1 for all (i,j) ∈ A×B

(ii) A and B are maximal with respect to (i), i.e. if A'⊇A and B'⊇B are such that (A',B') satisfies condition (i) then A=A' and B=B'

?

The answer is given by this expression.

Spoilers for what might possibly become a paper, but ...

Make an n×m matrix X where each entry X_{i,j}~Bernoulli(p) is chosen independently at random,

i.e. X_{i,j} = 1 with probability p and X_{i,j} = 0 with probability 1-p.

...

More honestly, I'd like to get some asymptotic control over this quantity as n,m -> infty

Nah, but it seems simple enough that I wouldn't be surprised if someone had thought about this sum before; maybe it's the expected value of some distribution people care about

oh!? if you could drop a link to something I would really appreciate it, I have no idea what those are :^)

and/or something like "this is the expected value of a Blorp(n,m,p)-distributed random variable" would be very helpful!

\sum_{i=0}^n \sum_{j=0}^m \binom{n}{i} \binom{m}{j} p^{i j} (1-p^i)^{m-j} (1-p^j)^{n-i}

... can this be simplified at all? n and m are fixed positive integers, p is a fixed real number between 0 and 1.

When I first learned about this I was baffled -- how can there possibly be only a set's worth of isomorphism classes of compact metric spaces???

But there is, and it's awesome

gl!

I love the metric space of isomorphism classes of compact metric spaces en.wikipedia.org/wiki/Gromov%...

More generally, we can ask: for which positive real numbers K can the inequality

|(f(z)-f(w))/(z-w)| ≤ K |f'(z)|

be satisfied?

K < 1 is impossible (consider f(z) = z^n - nz for arbitrary large integers n)

K ≥ 4 is possible (proved by Smale)

This is all we know!