## March 29, 2017

### Functional Equations VIII: Measuring Biodiversity

#### Posted by Tom Leinster

Posts entitled “Such-and-Such Part 8” can be intimidating…! But in this week’s instalment of the functional equations course, we just began a new topic. So if you’ve been interested in this course at a distance, without having actually dived in, now is a decent entry point.

This week, I introduced the difficult conceptual question of how to quantify biological diversity. A partial answer is given by what ecologists call the Hill numbers, which are the exponentials of what others call the Rényi entropies. I explained what these are, how they behave, and how they’re linked to other mathematical concepts.

There were hardly any actual functional equations this week, but they’ll come back soon! This week was mainly just background and intuition.

You can find the notes from this week’s session on pages 30–34 of the notes so far.

## March 22, 2017

### Functional Equations VII: The *p*-Norms

#### Posted by Tom Leinster

The $p$-norms have a nice multiplicativity property:

$\|(A x, A y, A z, B x, B y, B z)\|_p = \|(A, B)\|_p \, \|(x, y, z)\|_p$

for all $A, B, x, y, z \in \mathbb{R}$ — and similarly, of course, for any numbers of arguments.

Guillaume Aubrun and Ion Nechita showed that this condition completely characterizes the $p$-norms. In other words, *any* system of norms that’s multiplicative in this sense must be equal to $\|\cdot\|_p$ for some $p \in [1, \infty]$. And the amazing thing is, to prove this, they used some nontrivial probability theory.

All this is explained in this week’s functional equations notes, which start on page 26 here.

## March 21, 2017

### On the Operads of J. P. May

#### Posted by Emily Riehl

*Guest post by Simon Cho*

We continue the Kan Extension Seminar II with Max Kelly’s On the operads of J. P. May. As we will see, the main message of the paper is that (symmetric) operads enriched in a suitably nice category $\mathcal{V}$ arise naturally as monoids for a “substitution product” in the monoidal category $[\mathbf{P}, \mathcal{V}]$ (where $\mathbf{P}$ is a category that keeps track of the symmetry). Before we begin, I want to thank the organizers and participants of the Kan Extension Seminar (II) for the opportunity to read and discuss these nice papers with them.

## March 15, 2017

### Functional Equations VI: Using Probability Theory to Solve Functional Equations

#### Posted by Tom Leinster

A functional equation is an entirely deterministic thing, such as $f(x + y) = f(x) + f(y)$ or $f(f(f(x))) = x$ or $f\Bigl(\cos\bigl(e^{f(x)}\bigr)\Bigr) + 2x = \sin\bigl(f(x+1)\bigr).$ So it’s a genuine revelation that one can solve some functional equations using probability theory — more specifically, the theory of large deviations.

This week and next week, I’m explaining how. Today (pages 22-25 of these notes) was mainly background:

an introduction to the theory of large deviations;

an introduction to convex duality, which Simon has written about here before;

how the two can be combined to get a nontrivial formula for sums of powers of real numbers.

Next time, I’ll explain how this technique produces a startlingly simple characterization of the $p$-norms.

## March 10, 2017

### The Logic of Space

#### Posted by Mike Shulman

Mathieu Anel and Gabriel Catren are editing a book called *New Spaces for Mathematics and Physics*, about all different kinds of notions of “space” and their applications. Among other things, there are chapters about smooth spaces, $\infty$-groupoids, topos theory, stacks, and various other things of interest to $n$-Cafe patrons, all of which I am looking forward to reading. There are chapters by our own John Baez about the continuum and Urs Schreiber about higher prequantum geometry. Here is my own contribution:

### Postdocs in Sydney

#### Posted by Tom Leinster

*Richard Garner writes:*

The category theory group at Macquarie is currently advertising a two-year Postdoctoral Research Fellowship to work on a project entitled “Enriched categories: new applications in geometry and logic”.

Applications close 31st March. The position is expected to start in the second half of this year.

More information can be found at the following link:

http://jobs.mq.edu.au/cw/en/job/500525/postdoctoral-research-fellow

Feel free to contact me with further queries.

Richard Garner

## March 8, 2017

### Functional Equations V: Expected Surprise

#### Posted by Tom Leinster

In today’s class I explained the concept of “expected surprise”, which also made an appearance on this blog back in 2008: Entropy, Diversity and Cardinality (Part 1). Expected surprise is a way of interpreting the $q$-deformed entropies that I like to call “surprise entropies”, and that are usually and mistakenly attributed to Tsallis. These form a one-parameter family of deformations of ordinary Shannon entropy.

Also in this week’s session: $q$-logarithms, and a sweet, unexpected surprise:

Surprise entropies are much easier to characterize than ordinary entropy!

For instance, all characterization theorems for Shannon entropy involve some regularity condition (continuity or at least measurability), whereas each of its $q$-deformed cousins has an easy characterization that makes no regularity assumption at all.

It’s all on pages 18–21 of the course notes so far.

## March 7, 2017

### Algebra Valued Functors in General and Tensor Products in Particular

#### Posted by Emily Riehl

*Guest post by Maru Sarazola*

The Kan Extension Seminar II continues, and this time we focus on the article “Algebra valued functors in general and tensor products in particular” by Peter Freyd, published in 1966. Its purpose is to present algebraic theories and some related notions in a way that doesn’t make use of elements, so the concepts can later be applied to any category (satisfying some restrictions).

Concerned that the language of categories was not popular enough at the time, he chooses to target a wider audience by taking an “equational” approach in his exposition (in contrast, for example, to Lawvere’s more elegant approach, purely in terms of functors and natural transformations). I must say that this perspective, which nowadays might seem somewhat cumbersome, greatly helped solidify my understanding of some of these notions and constructions.

Before we start, I would like to thank Brendan Fong, Alexander Campbell and Emily Riehl for giving me the opportunity to take part in this great learning experience, and all the other participants for their enlightening comments and discussions. I would also like to thank my advisor, Inna Zakharevich, for her helpful comments and especially for her encouragement throughout this entire process.