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February 22, 2017

Functional Equations III: Explaining Relative Entropy

Posted by Tom Leinster

Much of this functional equations course is about entropy and its cousins, such as means, norms, and measures of diversity. So I thought it worth spending one session talking purely about ways of understanding entropy, without actually proving anything about it. I wanted especially to explain how to think about relative entropy — also known as relative information, information gain, and Kullback-Leibler divergence.

My strategy was to do this via coding theory. Information is a slippery concept, and reasoning about it takes some practice. But putting everything in the framework of coding makes everything more concrete. The central point is:

The entropy of a distribution is the mean number of bits per symbols in an optimal encoding.

All this and more is in the course notes. The part we did today starts on page 11.

Next week: relative entropy is the only quantity that satisfies a couple of reasonable properties.

Posted at 12:13 AM UTC | Permalink | Post a Comment

February 18, 2017

Distributive Laws

Posted by Emily Riehl

Guest post by Liang Ze Wong

The Kan Extension Seminar II continues and this week, we discuss Jon Beck’s “Distributive Laws”, which was published in 1969 in the proceedings of the Seminar on Triples and Categorical Homology Theory, LNM vol 80. In the previous Kan seminar post, Evangelia described the relationship between Lawvere theories and finitary monads, along with two ways of combining them (the sum and tensor) that are very natural for Lawvere theories but less so for monads. Distributive laws give us a way of composing monads to get another monad, and are more natural from the monad point of view.

Beck’s paper starts by defining and characterizing distributive laws. He then describes the category of algebras of the composite monad. Just as monads can be factored into adjunctions, he next shows how distributive laws between monads can be “factored” into a “distributive square” of adjunctions. Finally, he ends off with a series of examples.

Before we dive into the paper, I would like to thank Emily Riehl, Alexander Campbell and Brendan Fong for allowing me to be a part of this seminar, and the other participants for their wonderful virtual company. I would also like to thank my advisor James Zhang and his group for their insightful and encouraging comments as I was preparing for this seminar.

Posted at 7:06 AM UTC | Permalink | Followups (15)

February 14, 2017

Functional Equations II: Shannon Entropy

Posted by Tom Leinster

In the second instalment of the functional equations course that I’m teaching, I introduced Shannon entropy. I also showed that up to a constant factor, it’s uniquely characterized by a functional equation that it satisfies: the chain rule.

Notes for the course so far are here. For a quick summary of today’s session, read on.

Posted at 11:06 PM UTC | Permalink | Followups (12)

February 13, 2017

M-theory from the Superpoint

Posted by David Corfield

You may have been following the ‘Division algebra and supersymmetry’ story, the last instalment of which appeared a while ago under the title M-theory, Octonions and Tricategories. John (Baez) was telling us of some work by his former student John Huerta which relates these entities. The post ends with a declaration which does not suffer from comparison to Prospero’s in The Tempest

But this rough magic

I here abjure. And when I have required

Some heavenly music – which even now I do –

To work mine end upon their senses that

This airy charm is for, I’ll break my staff,

Bury it certain fathoms in the earth,

And deeper than did ever plummet sound

I’ll drown my book.

Posted at 1:21 PM UTC | Permalink | Followups (8)

February 10, 2017

The Heilbronn Institute and the University of Bristol

Posted by Tom Leinster

The Heilbronn Institute is the mathematical brand of the UK intelligence and spying agency GCHQ (Government Communications Headquarters). GCHQ is one of the country’s largest employers of mathematicians. And the Heilbronn Institute is now claiming to be the largest funder of “pure mathematics” in the country, largely through its many research fellowships at Bristol (where it’s based) and London.

In 2013, Edward Snowden leaked a massive archive of documents that shone a light on the hidden activities of GCHQ and its close partner, the US National Security Agency (NSA), including whole-population surveillance and deliberate stifling of peaceful activism. Much of this was carried out without the permission — or even knowledge — of the politicians who supposedly oversee them.

All this should obviously concern any mathematician with a soul, as I’ve argued. These are our major employers and funders. But you might wonder about the close-up picture. How do spy agencies such as GCHQ and the NSA work their way into academic culture? What do they do to ensure a continuing supply of mathematicians to employ, despite the suspicion with which most of us view them?

Alon Aviram of the Bristol Cable has just published an article on this, describing specific connections between GCHQ/Heilbronn and the University of Bristol — and, more broadly, academic mathematicians and computer scientists:

Alon Aviram, Bristol University working with the surveillance state. The Bristol Cable, 7 February 2017.

It includes some quotes from me and from legendary computer-security scientist Ross Anderson, as well as some nuggets from a long leaked Heilbronn “problem book” that’s interesting in its own right.

Posted at 2:42 PM UTC | Permalink | Post a Comment

February 7, 2017

Functional Equations I: Cauchy’s Equation

Posted by Tom Leinster

This semester, I’m teaching a seminar course on functional equations. Why? Among other reasons:

  1. Because I’m interested in measures of biological diversity. Dozens (or even hundreds?) of diversity measures have been proposed, but it would be a big step forward to have theorems of the form: “If you want your measure to have this property, this property, and this property, then it must be that measure. No other will do.”

  2. Because teaching a course on functional equations will force me to learn about functional equations.

  3. Because it touches on lots of mathematically interesting topics, such as entropy of various kinds and the theory of large deviations.

Today was a warm-up, focusing on Cauchy’s functional equation: which functions f:f: \mathbb{R} \to \mathbb{R} satisfy

f(x+y)=f(x)+f(y)x,y? f(x + y) = f(x) + f(y) \,\,\,\, \forall x, y \in \mathbb{R}?

(I wrote about this equation before when I discovered that one of the main references is in Esperanto.) Later classes will look at entropy, means, norms, diversity measures, and a newish probabilistic method for solving functional equations.

Read on for today’s notes and an outline of the whole course.

Posted at 11:25 PM UTC | Permalink | Followups (8)

The Category Theoretic Understanding of Universal Algebra

Posted by Emily Riehl

Guest post by Evangelia Aleiferi

We begin the second series of the Kan Extension Seminar by discussing the paper The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads by Martin Hyland and John Power, published in 2007. The subject of the above is to give a historical survey on the two main category theoretic formulations of universal algebra, the first being Lawvere theories and the second the theory of monads. Lawvere theories were introduced by William Lawvere in 1963 as part of his doctoral thesis, in order to provide an elegant categorical base for studying universal algebra, while monads were structures that had been already used in different areas of mathematics.

The article starts with the definition of a Lawvere theory and the category of its models, together with some of their properties. Later the authors proceed to relate the notion of monad to Lawvere theories and they give an explanation as to why they were dominantly used in the understanding of universal algebra, compared to Lawvere theories. Lastly they describe how monads and Lawvere theories can be used in formulating computational effects, motivated by the work of Moggi and Plotkin, and they propose future developments based on the connection between computational effects and universal algebra.

At this point, I would like to take the opportunity to thank Emily Riehl, Alexander Campbell and Brendan Fong for organizing the Kan Extension Seminar, as well as all the other participants for being such a great motivation!

Posted at 5:45 AM UTC | Permalink | Followups (31)

January 31, 2017

The Brunn-Minkowski Inequality

Posted by Tom Leinster

Let AA and BB be measurable subsets of n\mathbb{R}^n. Then

Vol(A+B) 1/nVol(A) 1/n+Vol(B) 1/n Vol(A + B)^{1/n} \geq Vol(A)^{1/n} + Vol(B)^{1/n}

where A+B={a+b:aA,bB}A + B = \{ a + b : a \in A, b \in B \}. This is the Brunn-Minkoswki inequality. It’s most often mentioned in the case where AA and BB are convex, but it’s true in the vast generality of measurable sets (except for \emptyset, where it obviously fails).

Here I want to explain just a few basic things about this inequality and its consequences. For instance, it easily implies the famous isoperimetric inequality: that among all sets with a given surface area, the one that maximizes the volume is the ball.

Posted at 10:48 PM UTC | Permalink | Followups (8)

January 24, 2017

Papers Written While Drunk

Posted by Tom Leinster

I’m currently reading a preprint by a deservedly very well-respected and highly-reputed mathematician. It’s enjoyable, inspirational, and wonderful. The ideas that it expresses have been haunting and taunting me for years.

For various reasons, I have the impression that it was not wholly written while the author was wholly sober. That’s OK; I’ll judge the paper for what it is, not on how it was written. But it leads me to wonder: how common is this? In literature, it’s a well-established tradition to the point of cliché. For instance, here’s Ernest Hemingway —

Hemingway in Cuba

— giving a cocktail recipe for difficult political times (1937), “to be enjoyed from 11:00am on”. You can find countless examples of fiction writers enthusing about chemically-assisted escape from the so-called real world.

But mathematics prides itself on sharpness and precision in counterpoint to creativity. We love to say that we’re more creative than poets, but a piece of mathematics is in deep trouble if it’s logically wrong. So where does drugged, drunk or hallucinatory mathematics fit into our mathematicians’ culture?

Posted at 11:44 PM UTC | Permalink | Followups (20)

January 10, 2017

Category Theory in Barcelona

Posted by Tom Leinster

I’m excited to be in Barcelona to help Joachim Kock teach an introductory course on category theory. (That’s a link to bgsmath.cat — categorical activities in Catalonia have the added charm of a .cat web address.) We have a wide audience of PhD and masters students, specializing in subjects from topology to operator algebras to number theory, and representing three Barcelona universities.

We’re taking it at a brisk pace. First of all we’re working through my textbook, at a rate of one chapter a day, for six days spread over two weeks. Then we’re going to spend a week on more advanced topics. Today Joachim did Chapter 1 (categories, functors and natural transformations), and tomorrow I’ll do Chapter 2 (adjunctions).

I’d like to use this post for two things: to invite questions and participation from the audience, and to collect slogans. Let me explain…

Posted at 6:48 PM UTC | Permalink | Followups (81)

January 4, 2017

Globular for Higher-Dimensional Knottings (Part 3)

Posted by John Baez

guest post by Scott Carter

This is my 3rd post a Jamie Vicary’s program Globular. And here I want to give you an exercise in manipulating a sphere in 4-dimensional space until it is demonstrably unknotted. But first I’ll need to remind you a lot about knotting phenomena. By the way, I lied. In the previous post, I said that the next one would be about braiding. I will write the surface braid post soon, but first I want to give you a fun exercise.

This post, then, will describe a 2-sphere embedded in 4-space, and we’ll learn to try and unknot it.

Posted at 6:08 AM UTC | Permalink | Followups (9)

January 2, 2017

Basic Category Theory Free Online

Posted by Tom Leinster

My textbook Basic Category Theory, published by Cambridge University Press, is now also available free as arXiv:1612.09375.

Cover of Basic Category Theory

As I wrote when I first announced the book:

  • It doesn’t assume much.
  • It sticks to the basics.
  • It’s short.

I can now add a new property:

  • It’s free.

And it’s not only free, it’s freely editable. The book’s released under a Creative Commons licence that allows you to edit and redistribute it, just as long as you state the authorship accurately, don’t use it for commercial purposes, and preserve the licence. Click the link for details.

Posted at 5:43 AM UTC | Permalink | Followups (21)

December 31, 2016

NSA Axes Math Grants

Posted by Tom Leinster

Old news, but interesting: the US National Security Agency (NSA) announced some months ago that it was suspending funding to its Mathematical Sciences Program. The announcement begins by phrasing it as a temporary suspension—

…[we] will be unable to fund any new proposals during FY2017 (i.e. Oct. 1, 2016–Sept. 30, 2017)

—but by the end, sounds resigned to a more permanent fate:

We thank the mathematics community and especially the American Mathematical Society for its interest and support over the years.

We’ve discussed this grant programme before on this blog.

The NSA is said to be the largest employer of mathematicians in the world, and has been under political pressure for obvious reasons over the last few years, so it’s interesting that it cut this programme. Its British equivalent, GCHQ, is doing the opposite, expanding its mathematics grants aggressively. But still, GCHQ consistently refuses to engage in any kind of adult, evidence-based discussion with the mathematical community on what the effect of its actions on society might actually be.

Posted at 3:39 AM UTC | Permalink | Followups (2)

December 17, 2016

Globular for Higher-Dimensional Knottings (Part 2)

Posted by John Baez

guest post by Scott Carter

This is the second post in a series about Globular. To load Globular, open a new tab in a Chrome browser window and have a a 3-button mouse plugged into your computer. The papers to read about Globular are Data structures for quasistrict higher categories by Jamie Vicary and Krzysztof Bar, and Globular: an online proof assistant for higher-dimensional rewriting in which Aleks Kissinger joins Jamie and Krzysztof to explain further and give some nice examples of globular’s potential.

Posted at 2:15 AM UTC | Permalink | Followups (14)

December 16, 2016

Globular for Higher-Dimensional Knottings

Posted by John Baez

guest post by Scott Carter

About 7 months ago, Jamie Vicary contacted me with a Globular worksheet of which, initially, I could make neither heads nor tails. He patiently explained to me that what I was looking at was an example that I had worked out for Bruce Bartlett one evening. He explained how to read it. Fast forward through a number of late night (for him) Skype sessions and a number of heartbreaking system errors for me, and now I feel that Globular is not only the best way to do higher dimensional knot theory and diagrammatic calculations, but it has the potential to be revolutionary. It will give insight into classical theorems and it will be used in the near future to create diagrammatic proofs of new theorems.

This first post will be about ordinary 3-dimensional knots.

Posted at 6:53 AM UTC | Permalink | Followups (5)