Ben Spitz
@diracdeltafunk.bsky.social
Sheaf Herder. I believe in you 🔥
benspitz.com
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Ben Spitz
@diracdeltafunk.bsky.social
· Sep 17
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.
The image below shows how you can use the package to compute a free resolution of a C_p-Mackey functor.
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.
The image below shows how you can use the package to compute a free resolution of a C_p-Mackey functor.
September 17, 2025 at 3:44 AM
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.
The image below shows how you can use the package to compute a free resolution of a C_p-Mackey functor.
... can this be simplified at all? n and m are fixed positive integers, p is a fixed real number between 0 and 1.
September 9, 2025 at 10:45 PM
... can this be simplified at all? n and m are fixed positive integers, p is a fixed real number between 0 and 1.
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)|?
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)|?
August 25, 2025 at 3:15 PM
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)|?
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)|?
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.
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.
August 23, 2025 at 1:45 PM
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.
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.
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)
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)
August 21, 2025 at 1:05 PM
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)
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)
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 ⊆?
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 ⊆?
August 20, 2025 at 2:49 PM
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 ⊆?
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 ⊆?
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)?
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)?
August 19, 2025 at 3:29 PM
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)?
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)?
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
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
August 18, 2025 at 3:46 PM
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
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
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...
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...
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...
arxiv.org
August 17, 2025 at 2:41 PM
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...
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...
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.
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.
August 16, 2025 at 3:36 PM
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.
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.
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|.
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|.
August 15, 2025 at 2:32 PM
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|.
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|.
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 ~
Must there exist a vector x ∈ Fⁿ such that neither x nor Ax has 0 as one of its entries?
~ spoilers below ~
August 14, 2025 at 10:24 PM
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 ~
Must there exist a vector x ∈ Fⁿ such that neither x nor Ax has 0 as one of its entries?
~ spoilers below ~
Fresh off the presses, joint with Scott Balchin! Come check out our fun pictures :)
arxiv.org/abs/2507.14068
arxiv.org/abs/2507.14068
July 21, 2025 at 1:06 PM
Fresh off the presses, joint with Scott Balchin! Come check out our fun pictures :)
arxiv.org/abs/2507.14068
arxiv.org/abs/2507.14068
Sneak research preview -- fun pictures coming soon
In this paper, we also get to take a limit as p (a prime) approaches 1 :^)
In this paper, we also get to take a limit as p (a prime) approaches 1 :^)
June 13, 2025 at 2:41 PM
Sneak research preview -- fun pictures coming soon
In this paper, we also get to take a limit as p (a prime) approaches 1 :^)
In this paper, we also get to take a limit as p (a prime) approaches 1 :^)
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" :)
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" :)
On Friday the 13th, a great result and excellent typography from the team that brought you "the ghost map" on Halloween. By 👻 Maxine Calle, 🎃 David Chan, 🦇 David Mehrle, 💀 JD Quigley, 👻 @diracdeltafunk.bsky.social, and 🧹 Danika Van Niel.
arxiv.org/abs/2506.10727
arxiv.org/abs/2506.10727
June 13, 2025 at 2:32 PM
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" :)
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" :)
I'm in Seattle for JMM! Hmu if you wanna grab a coffee or something :)
January 8, 2025 at 6:19 PM
I'm in Seattle for JMM! Hmu if you wanna grab a coffee or something :)
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?...
Title: "Algebraically Closed Tambara Functors"
youtu.be/ast_KRMwBOQ?...
Ben Spitz (Virginia) on "Algebraically Closed Tambara Functors"
YouTube video by eCHT
youtu.be
November 8, 2024 at 12:14 AM
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?...
Title: "Algebraically Closed Tambara Functors"
youtu.be/ast_KRMwBOQ?...
Check out our paper if you like Mackey and/or Tambara functors! It's spooky and fun
Seasonally appropriate arXiv posting: 'On the Tambara affine line' studies Tambara functors via a well-notated ghost map and the hyperlinks are orange 👻🎃 arxiv.org/abs/2410.23052
October 31, 2024 at 2:18 PM
Check out our paper if you like Mackey and/or Tambara functors! It's spooky and fun
Reposted by Ben Spitz
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
Conversely, if your hearing is such that you can understand everyone, it is called cochlear
October 18, 2024 at 2:34 PM
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
Conversely, if your hearing is such that you can understand everyone, it is called cochlear
Just made it to Indianapolis for #MathFest24! DM me if you're around and want to grab a coffee :)
August 6, 2024 at 11:33 PM
Just made it to Indianapolis for #MathFest24! DM me if you're around and want to grab a coffee :)
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
Feat. Talon on guitar, me on piano, Rohan Joshi on the board, and a great crowd of friends on vocals :)
youtu.be/xanZ3tAkhRo
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 ...
youtu.be
June 28, 2024 at 4:24 AM
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
Feat. Talon on guitar, me on piano, Rohan Joshi on the board, and a great crowd of friends on vocals :)
youtu.be/xanZ3tAkhRo
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/KvYYpJARP6M
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.
youtu.be
June 14, 2024 at 10:27 PM
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/KvYYpJARP6M
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
My intention was to make the exposition accessible to ~any math grad student; take a look if you're interested!
youtu.be/H5PsXY0E7Bw
Using Combinatorics to Compute Cohomology
This is a recording of a talk I gave for the UCLA Graduate Student Seminar, May 9 2024.
youtu.be
May 11, 2024 at 3:45 AM
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
My intention was to make the exposition accessible to ~any math grad student; take a look if you're interested!
youtu.be/H5PsXY0E7Bw
On my way to JMM! If you're in SF this week and wanna grab a coffee or something, hit me up :)
January 2, 2024 at 5:22 PM
On my way to JMM! If you're in SF this week and wanna grab a coffee or something, hit me up :)