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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 · 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

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 · 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

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 · 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 · 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

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

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

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

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 · 18/08/2025

An open problem in algebraic geometry: Let n be a positive integer. Let f : ℂⁿ → ℂⁿ be a regular* function such that the determinant of its Jacobian matrix is a nonzero constant. Must f be bijective with regular inverse? *i.e. each component ℂⁿ → ℂ is a polynomial

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Ben Spitz @diracdeltafunk.bsky.social · 17/08/2025

A crazy open problem in algebra: Conjecture (Poonen): 100% (asymptotic density) of finite commutative rings have characteristic EXACTLY 8, i.e. 4 ≠ 0 and 8 = 0 both hold. From this amazing paper: arxiv.org/abs/math/060...

arxiv.org

The moduli space of commutative algebras of finite rank

The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. I...

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Ben Spitz @diracdeltafunk.bsky.social · 16/08/2025

Another fun open problem: Conjecture (Rota). Let n be a natural number. Let V be an n-dimensional vector space. Let B₁, …, Bₙ be bases of V. Then there exist orderings of these bases Bₖ = (bₖ₁, …, bₖₙ) such that {b₁ₖ, …, bₙₖ} is a basis of V for all 1 ≤ k ≤ n.

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Ben Spitz @diracdeltafunk.bsky.social · 15/08/2025

My favorite open problem: Conjecture (Frankl). Let X be a finite set, and let S ⊆ P(X) be a collection of subsets of X which is closed under union. If S≠∅ and S≠{∅}, then some element x∈X appears in at least half of the elements of S, i.e. 2|{s ∈ S : x ∈ s}| ≥ |S|.

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Ben Spitz @diracdeltafunk.bsky.social · 14/08/2025

Let A be an n×n invertible matrix with coefficients in some field F. Must there exist a vector x ∈ Fⁿ such that neither x nor Ax has 0 as one of its entries? ~ spoilers below ~

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Ben Spitz @diracdeltafunk.bsky.social · 13/06/2025

Sneak research preview -- fun pictures coming soon In this paper, we also get to take a limit as p (a prime) approaches 1 :^)

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Ben Spitz @diracdeltafunk.bsky.social · 13/06/2025

Check out our paper for more spooky fun! Maxine and Sam Ginnett worked out this story for cyclic groups a few years ago, and together we realized that the ideas from our ghost paper this October might allow us to push the argument through for all finite groups. "Might" turned out to be "do" :)

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Ben Spitz @diracdeltafunk.bsky.social · 08/01/2025

I'm in Seattle for JMM! Hmu if you wanna grab a coffee or something :)

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Ben Spitz @diracdeltafunk.bsky.social · 08/11/2024

I gave a talk this morning on some recent research (joint with Jason Schuchardt and Noah Wisdom). If you like abstract algebra (broadly construed) you might like this one! Title: "Algebraically Closed Tambara Functors" youtu.be/ast_KRMwBOQ?...

youtu.be

Ben Spitz (Virginia) on "Algebraically Closed Tambara Functors"

YouTube video by eCHT

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Ben Spitz @diracdeltafunk.bsky.social · 31/10/2024

Check out our paper if you like Mackey and/or Tambara functors! It's spooky and fun

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Reposted by Ben Spitz

🇵🇸Tim Henke (tɪm 'ɦɛŋ.kə) @timhenke.bsky.social · 18/10/2024

If someone's diction is such that everyone can understand them, it is clear Conversely, if your hearing is such that you can understand everyone, it is called cochlear

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Ben Spitz @diracdeltafunk.bsky.social · 07/08/2024

Just made it to Indianapolis for #MathFest24! DM me if you're around and want to grab a coffee :)

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Ben Spitz @diracdeltafunk.bsky.social · 28/06/2024

A few months ago, Advika Rajapakse, Talon Stark, and I wrote a math parody(?) cover of "Imagine". Last weekend we played it at a party as a singalong! Feat. Talon on guitar, me on piano, Rohan Joshi on the board, and a great crowd of friends on vocals :) youtu.be/xanZ3tAkhRo

youtu.be

Imagine (There's no Zero)

Lyrics by Advika Rajapakse, Ben Spitz, and Talon StarkThis performance features Talon Stark on guitar, myself on piano, Rohan Joshi on the whiteboard, and a ...

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Ben Spitz @diracdeltafunk.bsky.social · 14/06/2024

I've started recording a short course on the representation theory of finite groups -- the first video is now up! Covering some basic motivation and a sneak peak of some cool facts we'll be able to prove by the end. youtu.be/KvYYpJARP6M

youtu.be

Intro to Rep Theory: Motivation

This video is a quick discussion of why we might care at all about the representation theory of finite groups.

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Ben Spitz @diracdeltafunk.bsky.social · 11/05/2024

I gave a talk yesterday on algebraic varieties, the Weil conjectures, and a strange application of the Grothendieck-Lefschetz trace formula. My intention was to make the exposition accessible to ~any math grad student; take a look if you're interested! youtu.be/H5PsXY0E7Bw

youtu.be

Using Combinatorics to Compute Cohomology

This is a recording of a talk I gave for the UCLA Graduate Student Seminar, May 9 2024.

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Ben Spitz @diracdeltafunk.bsky.social · 02/01/2024

On my way to JMM! If you're in SF this week and wanna grab a coffee or something, hit me up :)

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Ben Spitz @diracdeltafunk.bsky.social · 24/12/2023

Let M and N be smooth manifolds such that M and N have isomorphic groups of auto-diffeomorphisms. Must M and N be diffeomorphic?

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Ben Spitz @diracdeltafunk.bsky.social · 24/12/2023

The answer is: yes! For each d, you can take any finite coproduct of K_{d+1}'s. However, here's an interesting result of Biggs and Smith: there are exactly 12 finite connected 3-regular distance-transitive graphs, up to isomorphism.

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Ben Spitz @diracdeltafunk.bsky.social · 14/12/2023

A graph G is said to be distance-transitive if, for all vertices v,w,v',w', if d(v,w) = d(v',w'), then there is an automorphism φ of G such that φ(v) = v' and φ(w) = w'. Are there infinitely many finite d-regular distance-transitive graphs for all natural numbers d?

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Ben Spitz @diracdeltafunk.bsky.social · 13/12/2023

The answer is: yes! Such a set S is called a Steiner (n,k,m)-system. It turns out there is a unique Steiner (4,5,11)-system up to isomorphism! It has 66 elements, and can be found here: mathoverflow.net/a/452571/54637

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Ben Spitz @diracdeltafunk.bsky.social · 11/12/2023

A positive integer n is said to be "organizable" if there exist integers m > k > n and a set S ⊆ 2^{1,...,m} such that (i) each element of S has cardinality k (ii) for each X ⊆ {1,...,m} of cardinality n, there is a unique Y ∈ S such that X ⊆ Y Is 4 organizable?

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Ben Spitz @diracdeltafunk.bsky.social · 11/12/2023

The answer is: yes! First, let p be an arbitrary prime ideal of R. Then R/p is a finite domain, and hence a field. Thus, p is maximal. Also R is finite, so it has finitely many ideals. Thus, Spec(R) consists of finitely many closed points, so Spec(R) is finite and discrete.

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Ben Spitz @diracdeltafunk.bsky.social · 08/12/2023

Let R be a finite commutative ring. Must R be a product of local rings?

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Ben Spitz @diracdeltafunk.bsky.social · 08/12/2023

The answer is: no! I originally saw this posed on twitter by @chmonke, which sniped my friend Clark and I for a few days. You can read about our solution here twitter.com/DiracDeltaFunk… Quick counterexample: R = ℤ M = A⊕B with A, B the Prüfer 2- and 3-groups, resp. f = (a,b)↦(0,b)

twitter.com

@DiracDeltaFunk… on Twitter

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Ben Spitz @diracdeltafunk.bsky.social · 05/12/2023

Let R be a commutative ring, and let M be an R-module. Suppose f is an endomorphism of M with the following property: For all m ∈ M, there exists r ∈ R such that f(m) = rm. Must there exist r ∈ R such that for all m ∈ M, f(m) = rm?

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Ben Spitz @diracdeltafunk.bsky.social · 05/12/2023

The answer is: no! Proof: Let G be a group with Aut(G) ≅ ℤ/7. Then Inn(G) ≅ G/Z(G) is cyclic, so G is abelian. Now x ↦ -x : G → G is an automorphism of order ≤2. The only element of ℤ/7 of order ≤2 is the identity, so in fact x = -x for all x ∈ G. ...

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Ben Spitz @diracdeltafunk.bsky.social · 04/12/2023

Puzzle of the Day Does there exist a group G with Aut(G) ≅ ℤ/7?

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Ben Spitz @diracdeltafunk.bsky.social · 04/12/2023

The answer is: yes! This was asked by "mathlander" on math.stackexchange in 2022; I wrote a proof here: math.stackexchange.com/a/4587423/19006

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Ben Spitz @diracdeltafunk.bsky.social · 03/12/2023

Let C denote the ℝ-algebra of smooth functions ℝ → ℝ. Let d : C → C be a nonzero linear operator s.t. (i) d(f*g) = d(f)*g + f*d(g) (ii) d(f∘g) = (d(f)∘g)*d(g) i.e. d≠0 satisfies the product rule and the chain rule. Must d(f) = f' for all f ∈ C?

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Ben Spitz @diracdeltafunk.bsky.social · 03/12/2023

The answer is no! For example, we can take Y to be a the one-point space, and X any nondiscrete space. Then Hom(X,Y) is a singleton, which must be discrete.

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Ben Spitz @diracdeltafunk.bsky.social · 01/12/2023

Let X and Y be nonempty topological spaces such that Hom(X,Y) (with the compact-open topology) is discrete. Must X and Y be discrete?

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Ben Spitz @diracdeltafunk.bsky.social · 01/12/2023

The answer is: yes! Let χ be a character of a finite group G. Then ker(χ) := {g ∈ G : χ(g) = χ(1)} is a normal subgroup of G (equal to the kernel of the corresponding representation of G). Conversely, if N is a normal subgroup of G, then G acts on ℂ[G/N] with kernel N.

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Ben Spitz @diracdeltafunk.bsky.social · 30/11/2023

Suppose G and H are finite groups with the same character table. Suppose G is simple. Must H be simple?

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Ben Spitz @diracdeltafunk.bsky.social · 30/11/2023

The answer is: no! For example, take the ring of upper-triangular 2×2 matrices with integer coefficients. This ring is not commutative; in particular [2 0] [0 1] and [1 1] [0 1] do not commute.

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Ben Spitz @diracdeltafunk.bsky.social · 29/11/2023

Let R be a ring (unital, associative, not necessarily commutative) with underlying additive group ℤ³. Must R be commutative?

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