## December 16, 2017

### Entropy Modulo a Prime (Continued)

#### Posted by Tom Leinster

In the comments last time, a conversation got going about *$p$-adic*
entropy. But here I’ll return to the original subject: entropy *modulo
$p$*. I’ll answer the question:

Given a “probability distribution” mod $p$, that is, a tuple $\pi = (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n$ summing to $1$, what is the right definition of its entropy $H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}?$

## December 14, 2017

### Entropy Modulo a Prime

#### Posted by Tom Leinster

In 1995, the German geometer Friedrich Hirzebruch retired, and a private booklet was put together to mark the occasion. That booklet included a short note by Maxim Kontsevich entitled “The $1\tfrac{1}{2}$-logarithm”.

Kontsevich’s note didn’t become publicly available until five years later, when it was included as an appendix to a paper on polylogarithms by Philippe Elbaz-Vincent and Herbert Gangl. Towards the end, it contains the following provocative words:

Conclusion:If we have a random variable $\xi$ which takes finitely many values with all probabilities in $\mathbb{Q}$ then we can define not only the transcendental number $H(\xi)$ but also its “residues modulo $p$” for almost all primes $p$ !

Kontsevich’s note was very short and omitted many details. I’ll put some flesh on those bones, showing how to make sense of the sentence above, and much more.

## December 11, 2017

### The Icosahedron and E8

#### Posted by John Baez

Here’s a draft of a little thing I’m writing for the *Newsletter of the London Mathematical Society*. The regular icosahedron is connected to many ‘exceptional objects’ in mathematics, and here I describe two ways of using it to construct $\mathrm{E}_8$. One uses a subring of the quaternions called the ‘icosians’, while the other uses Du Val’s work on the resolution of Kleinian singularities. I leave it as a challenge to find the connection between these two constructions!

(Dedicated readers of this blog may recall that I was struggling with the second construction in July. David Speyer helped me a lot, but I got distracted by other work and the discussion fizzled. Now I’ve made more progress… but I’ve realized that the details would never fit in the *Newsletter*, so I’m afraid anyone interested will have to wait a bit longer.)

You can get a PDF version here:

• From the icosahedron to E_{8}.

But blogs are more fun.

## December 4, 2017

### The 2-Dialectica Construction: A Definition in Search of Examples

#### Posted by Mike Shulman

An adjunction is a pair of functors $f:A\to B$ and $g:B\to A$ along with a natural *isomorphism*

$A(a,g b) \cong B(f a,b).$

**Question 1:** Do we get any interesting things if we replace “isomorphism” in this definition by something else?

- If we replace it by “function”, then the Yoneda lemma tells us we get just a natural transformation $f g \to 1_B$.
- If we replace it by “retraction” then we get a unit and counit, as in an adjunction, satisfying one triangle identity but not the other.
- If $A$ and $B$ are 2-categories and we replace it by “equivalence”, we get a biadjunction.
- If $A$ and $B$ are 2-categories and we replace it by “adjunction”, we get a sort of lax 2-adjunction (a.k.a. “local adjunction”)

Are there other examples?

**Question 2:** What if we do the same thing for multivariable adjunctions?

A two-variable adjunction is a triple of functors $f:A\times B\to C$ and $g:A^{op}\times C\to B$ and $h:B^{op}\times C\to A$ along with natural isomorphisms

$C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)).$

What does it mean to “replace ‘isomorphism’ by something else” here? It could mean different things, but one thing it might mean is to ask instead for a *function*

$A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c).$

Even more intriguingly, if $A,B,C$ are 2-categories, we could ask for an ordinary *two-variable adjunction* between these three hom-categories; this would give a certain notion of “lax two-variable 2-adjunction”. Question 2 is, are notions like this good for anything? Are there any natural examples?

Now, you may, instead, be wondering about

**Question 3:** In what sense is a function $A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c)$ a “replacement” for isomorphisms $C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c))$?

But that question, I can answer; it has to do with comparing the Chu construction and the Dialectica construction.