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Ben Spitz @diracdeltafunk.bsky.social · 28/04/2026

Missing from mathematical english: "Maximal" is to "Maximality" as "Maximum" is to ????

Ben Spitz @diracdeltafunk.bsky.social · 09/02/2026

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.

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Ben Spitz @diracdeltafunk.bsky.social · 09/02/2026

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.

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Ben Spitz @diracdeltafunk.bsky.social · 15/01/2026

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.

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Ben Spitz @diracdeltafunk.bsky.social · 03/01/2026

I've arrived in DC for JMM! DM me if you're around, I'd love to grab coffee etc :)

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Ben Spitz @diracdeltafunk.bsky.social · 01/01/2026

Haha I should've looked at your full name

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Ben Spitz @diracdeltafunk.bsky.social · 01/01/2026

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

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Ben Spitz @diracdeltafunk.bsky.social · 01/01/2026

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."

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Ben Spitz @diracdeltafunk.bsky.social · 14/12/2025

Literally true, check out my papers 😎 bsky.app/profile/moti...

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Ben Spitz @diracdeltafunk.bsky.social · 10/12/2025

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)

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Ben Spitz @diracdeltafunk.bsky.social · 10/12/2025

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.

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Ben Spitz @diracdeltafunk.bsky.social · 10/12/2025

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.

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.

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Ben Spitz @diracdeltafunk.bsky.social · 09/12/2025

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

leonschools.net

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Ben Spitz @diracdeltafunk.bsky.social · 04/12/2025

On the ωth day of Christmas my true love gave to me ω numbers natural ... And a partridge in a pear treeeeee

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Ben Spitz @diracdeltafunk.bsky.social · 17/09/2025

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.

github.com

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

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Ben Spitz @diracdeltafunk.bsky.social · 17/09/2025

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.

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Ben Spitz @diracdeltafunk.bsky.social · 13/09/2025

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

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Ben Spitz @diracdeltafunk.bsky.social · 10/09/2025

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.

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Ben Spitz @diracdeltafunk.bsky.social · 10/09/2025

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.

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Ben Spitz @diracdeltafunk.bsky.social · 10/09/2025

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. ...

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Ben Spitz @diracdeltafunk.bsky.social · 09/09/2025

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

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Ben Spitz @diracdeltafunk.bsky.social · 09/09/2025

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

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Ben Spitz @diracdeltafunk.bsky.social · 09/09/2025

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

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Ben Spitz @diracdeltafunk.bsky.social · 09/09/2025

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

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Ben Spitz @diracdeltafunk.bsky.social · 09/09/2025

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

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

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Ben Spitz @diracdeltafunk.bsky.social · 04/09/2025

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

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Ben Spitz @diracdeltafunk.bsky.social · 04/09/2025

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

en.wikipedia.org

Gromov–Hausdorff convergence - Wikipedia

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Ben Spitz @diracdeltafunk.bsky.social · 25/08/2025

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!

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Ben Spitz @diracdeltafunk.bsky.social · 25/08/2025

An open problem in complex analysis: Let f ∈ ℂ[x] be a polynomial of degree ≥2. Let z ∈ ℂ. Must there exist a critical point w of f such that |(f(z)-f(w))/(z-w)| ≤ |f'(z)|?

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Ben Spitz @diracdeltafunk.bsky.social · 23/08/2025

Yeah this is kind of unclear to me; I've seen this implied but I can't find a reference

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Ben Spitz @diracdeltafunk.bsky.social · 23/08/2025

There is a unique Moore graph of diameter 2 and degree 2 (C_5). There is a unique Moore graph of diameter 2 and degree 3 (the Petersen graph) There is a unique Moore graph of diameter 2 and degree 7 (see link) Is there a Moore graph of diameter 2 and degree 57?? We don't know!

en.wikipedia.org

Hoffman–Singleton graph - Wikipedia

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Ben Spitz @diracdeltafunk.bsky.social · 23/08/2025

Well ok, Moore graphs do exist: for example, the complete graphs K_n (for n≥3) and the odd cycle graphs C_{2n+1} (for n≥1). So it would be nice to classify them! Theorem (Hoffman-Singelton). Let G be a Moore graph of diameter 2. Then G has degree 2, 3, 7, or 57.

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Ben Spitz @diracdeltafunk.bsky.social · 23/08/2025

Theorem 1: Every Moore graph is regular. Theorem 2: Let G be a finite graph with diameter k. Then G is a Moore graph if and only if G has girth 2k+1. This is a nice characterization, but we should ask how common Moore graphs actually are — a priori, they might not exist at all!

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Ben Spitz @diracdeltafunk.bsky.social · 23/08/2025

This question might seem completely unmotivated, but bear with me! If G is a finite graph with maximum degree d and diameter k, you can show that G has at most 1 + d ∑_{i=0}^{k-1} (d-1)^i many vertices. Definition. A "Moore graph" is a finite graph which attains this bound.

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Ben Spitz @diracdeltafunk.bsky.social · 23/08/2025

An open question in graph theory: Does there exist a finite (simple, undirected) graph which has diameter 2, girth* 5, and is 57-regular? * The girth of a graph G is the smallest length of a cycle in G.

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Ben Spitz @diracdeltafunk.bsky.social · 21/08/2025

Note that an equivalent formulation of the conjecture is as follows: For each z ≥ 0, let A(z) = |{n ≥ 1 : ϕ(n) = z}|, so that A is a function ℕ → ℕ ∪ {ℵ₀}. Conjecture. 1 is not in the image of A. It is known that every natural number besides 1 is in the image of A!

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Ben Spitz @diracdeltafunk.bsky.social · 21/08/2025

This was originally published as a theorem by Carmichael (over 100 years ago), but his proof was wrong. And today it's still open! We know that if there is any counterexample x, it must satisfy x > 10^(10^10).

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Ben Spitz @diracdeltafunk.bsky.social · 21/08/2025

An open problem in number theory: Recall that the totient function ϕ is defined by sending each positive integer n to the number of positive integers k ≤ n which are coprime to n. Conjecture. For all positive integers x, there exists a positive integer y≠x such that ϕ(x)=ϕ(y)

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Ben Spitz @diracdeltafunk.bsky.social · 20/08/2025

One day society will move past the need for spectral sequences, but it is not yet that day

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Ben Spitz @diracdeltafunk.bsky.social · 20/08/2025

Ofc WLOG one can take K=G in the above... I dunno why I wrote it that way 🙃 en.wikipedia.org/wiki/Finite_...

en.wikipedia.org

Finite lattice representation problem - Wikipedia

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Ben Spitz @diracdeltafunk.bsky.social · 20/08/2025

This is equivalent to the "finite lattice representation problem", which asks if every finite lattice is isom. to the congruence lattice of some finite algebra (in the sense of universal algebra). In other words, is there any restriction on the cong. lattices of finite algebras?

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Ben Spitz @diracdeltafunk.bsky.social · 20/08/2025

An open problem in order theory: Let L be a finite lattice (i.e. a nonempty finite poset such that any two elements have both an inf and a sup). Must there exist a finite group G with subgroups H, K such that L is isomorphic to the poset {X ≤ G : H ≤ X ≤ K} ordered by ⊆?

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Ben Spitz @diracdeltafunk.bsky.social · 19/08/2025

Ah yes, the sequence of interesting numbers

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Ben Spitz @diracdeltafunk.bsky.social · 19/08/2025

For 3-term arithmetic progressions, the answer is "yes" (exercise, if you'd like!) For 5-term arithmetic progressions, the answer is "no" (eudml.org/doc/205594) This 4-term case remains open!

eudml.org

EUDML | On permutations containing no long arithmetic progressionsEuDML | On permutations containing no long arithmetic progressions

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Ben Spitz @diracdeltafunk.bsky.social · 19/08/2025

Another elementary open problem: Let f : ℕ → ℕ be a bijection. Must there exist a natural number n and a positive integer k such that either f(n) > f(n+k) > f(n+2k) > f(n+3k) or f(n) < f(n+k) < f(n+2k) < f(n+3k)?

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Ben Spitz @diracdeltafunk.bsky.social · 19/08/2025

My claim to fame is being the first person (I think) to find 4×4 examples of this

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