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December 16, 2017

Entropy Modulo a Prime (Continued)

Posted by Tom Leinster

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

Given a “probability distribution” mod pp, that is, a tuple π=(π 1,,π n)(/p) n \pi = (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n summing to 11, what is the right definition of its entropy H p(π)/p? H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}?

Posted at 4:53 PM UTC | Permalink | Post a Comment

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 1121\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 ξ\xi which takes finitely many values with all probabilities in \mathbb{Q} then we can define not only the transcendental number H(ξ)H(\xi) but also its “residues modulo pp” for almost all primes pp !

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.

Posted at 11:00 PM UTC | Permalink | Followups (9)

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 E 8 \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 E8.

But blogs are more fun.

Posted at 1:13 AM UTC | Permalink | Followups (14)

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:ABf:A\to B and g:BAg:B\to A along with a natural isomorphism

A(a,gb)B(fa,b). 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 fg1 Bf 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 AA and BB are 2-categories and we replace it by “equivalence”, we get a biadjunction.
  • If AA and BB 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×BCf:A\times B\to C and g:A op×CBg:A^{op}\times C\to B and h:B op×CAh:B^{op}\times C\to A along with natural isomorphisms

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

Even more intriguingly, if A,B,CA,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))×B(b,g(a,c))C(f(a,b),c) 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)B(b,g(a,c))A(a,h(b,c)) 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.

Posted at 10:26 PM UTC | Permalink | Followups (4)