Ben Spitz
@diracdeltafunk.bsky.social
Sheaf Herder. I believe in you 🔥
benspitz.com
benspitz.com
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/Macaulay2/M2...
I think this will genuinely save equivariant homotopy theorists a lot of time and hair-wringing, I'm so stoked.
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
github.com
September 17, 2025 at 3:44 AM
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/Macaulay2/M2...
I think this will genuinely save equivariant homotopy theorists a lot of time and hair-wringing, I'm so stoked.
"What can we do about this? Simply choose to live in the worst of both worlds."
September 13, 2025 at 4:42 PM
"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.
And more generally in the distribution of the number of such pairs (A,B), but that seems much harder than just studying the mean.
September 10, 2025 at 12:19 AM
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.
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.
(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.
September 10, 2025 at 12:18 AM
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.
(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.
...
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.
...
September 10, 2025 at 12:18 AM
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.
...
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
September 9, 2025 at 10:57 PM
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
September 9, 2025 at 10:56 PM
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 :^)
September 9, 2025 at 10:54 PM
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!
September 9, 2025 at 10:49 PM
and/or something like "this is the expected value of a Blorp(n,m,p)-distributed random variable" would be very helpful!
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
But there is, and it's awesome
September 4, 2025 at 9:18 PM
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
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%...
I love the metric space of isomorphism classes of compact metric spaces en.wikipedia.org/wiki/Gromov%...
Gromov–Hausdorff convergence - Wikipedia
en.wikipedia.org
September 4, 2025 at 8:36 PM
gl!
I love the metric space of isomorphism classes of compact metric spaces en.wikipedia.org/wiki/Gromov%...
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!
|(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!
August 25, 2025 at 3:15 PM
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!
|(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!
Yeah this is kind of unclear to me; I've seen this implied but I can't find a reference
August 23, 2025 at 3:03 PM
Yeah this is kind of unclear to me; I've seen this implied but I can't find a reference
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!
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!
Hoffman–Singleton graph - Wikipedia
en.wikipedia.org
August 23, 2025 at 1:45 PM
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!
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!
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.
Theorem (Hoffman-Singelton). Let G be a Moore graph of diameter 2. Then G has degree 2, 3, 7, or 57.
August 23, 2025 at 1:45 PM
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.
Theorem (Hoffman-Singelton). Let G be a Moore graph of diameter 2. Then G has degree 2, 3, 7, or 57.
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!
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!
August 23, 2025 at 1:45 PM
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!
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!
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.
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.
August 23, 2025 at 1:45 PM
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.
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.
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!
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!
August 21, 2025 at 1:05 PM
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!
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!
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).
We know that if there is any counterexample x, it must satisfy x > 10^(10^10).
August 21, 2025 at 1:05 PM
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).
We know that if there is any counterexample x, it must satisfy x > 10^(10^10).