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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.

Posted at 2:32 AM UTC | Permalink | Followups (4)

March 22, 2017

Functional Equations VII: The p-Norms

Posted by Tom Leinster

The pp-norms have a nice multiplicativity property:

(Ax,Ay,Az,Bx,By,Bz) p=(A,B) p(x,y,z) p \|(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,zA, 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 pp-norms. In other words, any system of norms that’s multiplicative in this sense must be equal to p\|\cdot\|_p for some p[1,]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.

Posted at 2:12 AM UTC | Permalink | Post a Comment

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 𝒱\mathcal{V} arise naturally as monoids for a “substitution product” in the monoidal category [P,𝒱][\mathbf{P}, \mathcal{V}] (where P\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.

Posted at 11:07 AM UTC | Permalink | Followups (3)

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) f(x + y) = f(x) + f(y) or f(f(f(x)))=x f(f(f(x))) = x or f(cos(e f(x)))+2x=sin(f(x+1)). 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 pp-norms.

Posted at 12:56 AM UTC | Permalink | Post a Comment

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 nn-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:

Posted at 12:58 PM UTC | Permalink | Followups (15)

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:

Feel free to contact me with further queries.

Richard Garner

Posted at 12:19 AM UTC | Permalink | Post a Comment

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 qq-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: qq-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 qq-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.

Posted at 1:36 AM UTC | Permalink | Followups (13)

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.

Posted at 10:22 AM UTC | Permalink | Followups (6)