Ben Spitz

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!

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)|?

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

Hoffman–Singleton graph - Wikipedia

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!

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 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!

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.

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.

en.wikipedia.org/wiki/Carmich...

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!

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).

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)

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

Ofc WLOG one can take K=G in the above... I dunno why I wrote it that way 🙃

en.wikipedia.org/wiki/Finite_...

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?

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 ⊆?

Ah yes, the sequence of interesting numbers