January 30, 2014

Weightings for Compact Metric Spaces

Posted by Tom Leinster

Guest post by Mark Meckes

A recent paper of Mark’s proved the very substantial result that Minkowski dimension (one of the most important types of fractal dimension) can be derived from magnitude (an invariant ultimately coming from category theory). More exactly, he showed that the Minkowski dimension of a compact subset of $\mathbb{R}^n$ is exactly equal to its “magnitude dimension” (Cor 7.4). Here, Mark explains not this specific result, but the overall framework that makes such theorems possible. —TL

I’d like to report on some recentish progress in understanding the magnitude of compact metric spaces. To set the stage I’ll recap some background, which will repeat a number of things which have already appeared in posts by Tom and Simon. For those who want to be reminded of more, Tom has provided a reading list.

Posted at 4:45 PM UTC | Permalink | Followups (17)

Tensor Categories and TQFT in Erlangen

Posted by John Baez

There’s a meeting in Erlangen coming up!

Structures on Tensor Categories and Topological Field Theories, Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, March 4-6, 2014, organized by Catherine Meusburger and Christoph Schweigert.

Posted at 11:33 AM UTC | Permalink | Followups (4)

January 27, 2014

Formal Theory of Monads (Following Street)

Posted by Emily Riehl

Guest post by Eduard Balzin

The Kan extension seminar continues, and with it we now come to the paper by Ross Street. Published in 1972, this paper is one of the first instances where the notion of a monad was made relative to an arbitrary 2-category. A lot of aspects, such as algebras over a monad, Eilenberg-Moore and Kleisli categories, the relation with adjunctions, were generalised accordingly. Other topics involve representability, duality, and the example of $\mathbf{Cat}$. I will therefore try to talk about this paper in detail, and add something of my own.

Myself, in the course of my (higher)-categorical education, I have not learned in full about monads. I do not mean definitions or main theorems, but rather the answer to the question why monads. While preparing for the seminar, I’ve managed to give a series of answers to this (my own) question, and I hope that the discussion will take me (and everyone else) even further.

Let me sincerely thank Emily Riehl for organising the seminar. I truly hope it will be a new castle on the landscape of categorical life. I am also quite grateful for the participants of the seminar, who have greatly contributed during the online discussion and in their readers discussion. It all makes me happy.

Posted at 6:53 PM UTC | Permalink | Followups (42)

January 22, 2014

The Magnitude of a Graph

Posted by Tom Leinster

I’ve just arXived a new paper about a new invariant, The magnitude of a graph. Much of the development of this invariant has taken place at this blog, with two previous posts and crucial contributions from David Speyer and Simon Willerton. Your comments, critical or otherwise, would be very welcome.

Magnitude seems to be orthogonal to other graph invariants. You can’t derive from it such classic invariants as the Tutte polynomial or the chromatic number, nor the girth nor the clique number nor even the number of connected components. And conversely, you can’t derive the magnitude from these or any other well-known graph invariants. Apparently, it captures information of a different kind.

Its most appealing characteristic is that it behaves like cardinality. It’s multiplicative with respect to cartesian product, additive with respect to disjoint union, and obeys a restricted version of the inclusion-exclusion principle. And it also behaves a bit like the Tutte polynomial. For instance, it’s often invariant under Whitney twists — though not always, as I’ll explain.

Posted at 12:19 AM UTC | Permalink | Followups (15)

January 21, 2014

Hilarious Takedown of Bonkers Maths in Top Psychology Journal

Posted by Tom Leinster

If someone told you that in order to flourish rather than languish, you have to have at least 2.9013 times as many positive as negative emotions, and that they know this because of the Lorenz equations, you’d be instantly suspicious, right? It sounds like run-of-the-mill pseudoscience, but it was published in American Psychologist, official organ of the American Psychological Association and one of the very top journals in the field. Not only that, the lead author on this two-author paper, Barbara Fredrickson, is one of the editors of this journal.

The Guardian Observer carried a great piece yesterday telling how it was debunked by — it gets better — a master’s student, Nick Brown. He teamed up with the legendary Alan Sokal and the psychologist Harris Friedman, and they wrote this highly enjoyable account of exactly how incompetent the original paper was. American Psychologist, to its credit, is publishing it.

Posted at 1:53 PM UTC | Permalink | Followups (19)

January 19, 2014

Wormholes and Entanglement

Posted by John Baez

For the last couple years, people interested in quantum gravity have been arguing about the ‘firewall problem’. It’s a thought experiment involving black holes that claims to demonstrate an inconsistency in some widely held assumptions about how quantum mechanics and general relativity fit together. Everyone is either scratching their heads over it, struggling to find a way out… or grumbling that the problem isn’t real, and everyone else has gone crazy.

While the firewall problem has roots going back earlier, the paper that got everyone interested came out in July 2012:

They called it the ‘firewall’ problem because one way out is to assume that when you fall into a black hole you hit a ‘wall of fire’ — a region of hot radiation — as you cross the event horizon. That sounds crazy: the equivalence principle says you shouldn’t feel anything special as you fall through the horizon, at least if the black hole is big enough. But they claimed this crazy-sounding solution was the most conservative way out!

In June of the following year, two extremely reputable physicists, in their attempts to avoid this conclusion, came up with an idea that sounds even crazier: every pair of entangled quantum particles is connected by a wormhole!

They called their proposal EPR = ER, which is a joke referring to two papers Einstein wrote in 1935. EPR stands for ‘Einstein–Podolsky–Rosen’, the authors of a famous paper on quantum entanglement, a spooky way that distant objects can be correlated. ER stands for ‘Einstein–Rosen’, the authors of a famous paper on wormholes, solutions of general relativity in which distant regions of space can be connected by a kind of tunnel, or handle.

I haven’t been thinking much about quantum gravity lately, but the recent convulsions in my old favorite subject caught my attention. In particular, I’d long been fond of the idea that when we finally understand quantum gravity, the distinction between quantum mechanics and our theory of spacetime will evaporate. There’s a lot of evidence coming from category theory, which reveals analogies between the two:

Now, quantum entanglement is a sneaky way for two distant particles to be correlated. A wormhole is a sneaky way for two distant particles to be connected. Could this be hinting at yet another analogy between quantum mechanics and our theory of spacetime? Perhaps one that deserves a category-theoretic treatment?

Last summer, when Jamie Vicary and I were both at the Centre for Quantum Technologies, we figured out something interesting about this. We wrote a paper about it, which is done now:

I’d like to explain a bit of it here, because it uses 2-categories in a cute way.

Posted at 10:37 AM UTC | Permalink | Followups (43)

January 13, 2014

The Electronic Frontier Foundation at the Joint Meetings

Posted by Tom Leinster

Going to the Joint Mathematics Meetings in Baltimore this week? Then drop in at Booth 330, which will be occupied by the Electronic Frontier Foundation.

The EFF have been doing fantastic work for over 20 years, keeping the internet the kind of place you most likely want it to be: ensuring your freedom of speech, protecting your privacy, and defending the core principles of the internet against the controlling ambitions of both corporations and governments. Although they’re only a small nonprofit organization, with a comparatively minuscule budget, they’ve had a string of legal victories against huge players. They deserve your support!

But what are they doing at the Joint Mathematics Meetings? It was the idea of Thomas Hales. Hales is famous for, among other things, proving the Kepler sphere-packing conjecture. (He also wrote a very nice introduction to motivic measure, mentioned here in passing a couple of years ago.)

Hales anticipated that the NSA would be recruiting mathematicians with particular fervour this year: in order to recruit, they’ll need to overcome the outrage caused by the recent revelations of mass, population-level, surveillance. They’ll want to persuade mathematicians that what they’re doing is good for society. And the EFF will be there to tell mathematicians that there may be better channels for their talents.

Posted at 2:58 AM UTC | Permalink | Followups (17)

January 12, 2014

An Elementary Theory of the Category of Sets

Posted by Emily Riehl

William Lawvere’s Elementary Theory of the Category of Sets (ETCS) was one of the first attempts at using category theory as a foundation of mathematics and formulating set theory in category theoretic language. In this post I will outline the content of Lawvere’s paper [Lawvere 1965] and put it in a historical context; suggestions for further reading are at the end. Before I begin, I’d like to thank Emily Riehl, Steve Awodey and the other Kan Extension Seminar participants for their helpful feedback, suggestions, and reading responses to the paper; and, of course, William Lawvere for writing it!

Posted at 6:31 PM UTC | Permalink | Followups (88)

January 8, 2014

Posted by Tom Leinster

I’ve posted before about the involvement of mathematicians and computer scientists in the secret services — who, we now know, routinely record your and my daily activities. And in case you’ve somehow missed this whole scandal, you could do a lot worse than starting with this piece by Bruce Schneier in The Atlantic.

Many academics oppose mass surveillance. This probably includes many mathematicians who have helped the NSA and GCHQ in the past. If you’re one of them, you can add your name to a public declaration: Academics Against Mass Surveillance. All you need to do is this:

email us at info (at) academicsagainstsurveillance.net with your name, academic function and university in the subject line.

Given their pivotal role, I hope that many more mathematicians will want to add their names. (There may be some in the pipeline, along with my own: apparently, the organizers can’t keep up with the volume of emails coming in, citing an “overwhelming response”.)

January 5, 2014

Ends

Posted by Simon Willerton

The categorical notion of an end is something that several people have requested Catster videos for and Yemon Choi was recently asking if Tom had covered it in his new-born book. Given that I’ve got ends in my head at the moment for two different reasons, I thought I’d write a post on how I think about them.

I feel that seeing an integral sign like $\int _{c\in \mathcal{C}} T(c,c)$ can cause people’s eyes to glaze over, never mind them getting confused as to whether that represents an end or a coend. So I will endeavour to avoid integral signs apart from right at the end.

My experience is that coends roam more freely in the wild than ends do, but I will focus on ends in this post. One reason that people are interested in ends is that natural transformation objects in enriched category theory are expressed as ends, but I will stay away from the enriched setting here.

Having said what I won’t do, maybe I should say what I will do. I will mainly concentrate on a few examples to demonstrate ends as universal wedges.

Posted at 6:07 PM UTC | Permalink | Followups (33)