## December 10, 2013

### A Technical Innovation

#### Posted by Tom Leinster

Here’s a new feature of the Café, thanks to our benevolent host Jacques Distler. If you ever want to see how someone has created some mathematical expression on this blog, there’s an easy way to do it.

With Firefox, you simply double-click on the expression. Try it: $A \times B^A \to B$ or $x_{m n}$ or

$\Biggl( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \Biggr).$

A window should pop up showing the TeX source.

With other browsers, I’m not so sure. Try double-clicking. If that doesn’t work, then, according to Jacques’s instructions, you “bring up the MathJax context-menu for the formula, and choose Show Math As $\to$ Annotation $\to$ TeX”. I don’t know how one brings up this menu. Does anyone else know?

Once you’ve made the TeX source appear, you can cut and paste to your heart’s content. Of course, most users here are fluent in LaTeX. But like most math-oriented websites, we use a variant of TeX that’s a little different from standard LaTeX, so this should turn out to be a helpful feature.

Posted at 1:41 AM UTC | Permalink | Followups (4)

## November 25, 2013

### Kan Extension Seminar applications

#### Posted by Emily Riehl

A friendly reminder: applications for the Kan Extension Seminar are due at the end of the week. More information can be found in the initial announcement and on the seminar website.

For those who don’t enroll, watch this space. You’ll be hearing from us again soon after the new year.

Posted at 5:04 PM UTC | Permalink | Followups (1)

## November 21, 2013

### The 8th Scottish Category Theory Seminar

#### Posted by Tom Leinster

ScotCats 8 will take place in Edinburgh next Friday, 29 November, with an all-star programme featuring one of your $n$-Café hosts:

I’m particularly looking forward to this one: it promises to be an excellent afternoon of talks.

The seminar web page has practical details. Thanks to the Glasgow Mathematical Journal Trust and the Scottish Informatics and Computer Science Alliance for funding.

## November 19, 2013

### The Covariance of Coloured Balls

#### Posted by David Corfield

When you reach over 150 messages in response to a post, it’s probably time to start a new one, especially for those with inefficient browsers. In this post I want to see if I can make sense of discussions we were having there about general relativity, covariance, groupoids and sameness in a much more intuitively clear setting, though perhaps it will introduce its own problems.

OK, so say I have a set, $B$, of 5 indiscernible balls in a box, each of which can be coloured in 3 ways. Then I might take the state space to be $[B, 3]$, of cardinality 243. But perhaps while doing physics on these balls, I come to see that nothing about them matters aside from their colour. Since I can’t tell the difference between any two versions of 2 reds, 2 greens and a blue, I decide to reduce my state space to $[B, 3]/Sym(B)$, a set of equivalence classes of cardinality 21.

However, after a time this starts to strike me as odd. I have represented the fact that any colouring of 2 reds, 2 greens and a blue are equal, and yet were someone to swap a green and a red I’d notice in a way that I wouldn’t notice a swap of two reds. Any swaps in a state of 5 red balls would also go unnoticed. At the very least, there seems to be a difference between a monochrome state and a tricoloured state.

Posted at 12:35 PM UTC | Permalink | Followups (136)

## November 18, 2013

### Categories for the Working Philosopher

#### Posted by John Baez

Elaine Landry, in the philosophy department at U. C. Davis, is putting together a book called Categories for the Working Philosopher.

Posted at 4:09 AM UTC | Permalink | Followups (51)

## November 12, 2013

### Four New Talks

#### Posted by Tom Leinster

In October I did little but talk. Five talks in five locations in 23 days, with only one duplicate among them, left me heartily wishing not to hear my own voice for a while.

Having gone to the effort of making slides, I might as well share them publicly. All the talks are on topics that have come up on the Café before. Here they are:

Posted at 4:02 PM UTC | Permalink | Followups (4)

## November 10, 2013

### Severing Ties with the NSA

#### Posted by Tom Leinster

A letter from Chicago mathematician Sasha Beilinson in this month’s Notices of the American Mathematical Society calls for the AMS to sever all ties with the US National Security Agency, citing

the vast secret spying programs of the NSA that wildly exceed anything conspiracy theorists could imagine.

He lists some of the ways in which the AMS and NSA support each other, and issues a call for action:

What should be done is a question not only for US citizens but also for people all over the world: the NSA destroyed the security of the Internet and privacy of communications for the whole planet. But if any healing is possible, it would probably start with making the NSA and its ilk socially unacceptable — just as, in the days of my youth, working for the KGB was socially unacceptable for many in the Soviet Union.

Posted at 7:23 PM UTC | Permalink | Followups (60)

## October 30, 2013

### The HoTT Approach to Physics

#### Posted by David Corfield

Summer saw the foundations of mathematics rocked by the publication of The HoTT Book. Here we are a few months later and the same has happened to physics with the appearance on the ArXiv of Urs’s Differential cohomology in a cohesive infinity-topos.

Physics clearly needs more than the bare homotopy types of HoTT. Field configurations may be groupoids (1-types) under gauge equivalence, or indeed $\infty-groupoids$ (homotopy types) under gauge-of-gauge-of-… equivalence, but they also possess differentiable structure. The question then is how to cater for all of those principal bundles, connections, curvature forms, and in more recent times 2-bundles, orbifolds, Lie $\infty$-algebroids,…, while building on HoTT, or, in terms of the environments in which HoTT (or Univalent Foundations) functions, while adding structure to $(\infty, 1)$-toposes.

Posted at 10:21 AM UTC | Permalink | Followups (159)

## October 21, 2013

### Jet Categories at the nForum

#### Posted by David Corfield

Some people I talk to who have noticed a slackening off at the Café in recent months, and who know that some of this is due to John’s energy passing to his Azimuth project, don’t seem aware that another chunk of the energy didn’t just vanish, but got transmitted to the nForum. This venue has the advantage of democratically allowing anyone to initiate a discussion, but the disadvantage that people don’t seem to want to wade through every announcement of any alteration to an nLab entry for the occasional interesting nugget. For whatever reason, we don’t get visited much nowadays by some of the prestigious visitors of yesteryear. Still, if you want to see the day to day movements of Urs sweeping up great clumps of mathematical physics into a glorious synthetic package, the nForum is the place to be. Or, if you prefer to read the finished product, see his site.

Something we occasionally suggest to each other is a periodic digest of what’s happening at the $n$Lab, but I believe we’ve only managed two to date (I, II). Let me try something less ambitious.

Posted at 10:23 AM UTC | Permalink | Followups (3)

## October 16, 2013

### Announcing the Kan Extension Seminar

#### Posted by Emily Riehl

Daniel Kan’s influence at MIT persists through something called the Kan seminar, a graduate reading course in algebraic topology. Over the course of a semester, each student is asked to give a few one-hour lectures summarizing classic papers in the field and to engage with each other paper by writing a reading response. The lectures are preceded by a practice talk of unbounded length that is conducted in private, i.e., in the absence of the lead instructor, before the reading responses are due. This format aims to teach students how to read papers quickly and at various levels of depth, as well as to work on presentation skills. At the semester’s conclusion, Kan traditionally hosted a party that took advantage of Boston’s high concentration of mathematicians, giving his students an opportunity to meet senior people in the field.

This (northern hemisphere) spring, from early January to late June 2014, I plan to run an online (“extension”) Kan seminar in category theory with the aim of reading the twelve papers listed below. I am seeking between 6 and 12 participants who will compose one or two blog posts to appear here on the $n$-Category Café over the course of the six months, which will be published every other week. Everyone will be expected to write comments, engaging with all of the papers.

• Lawvere, An elementary theory of the category of sets

• Street, The formal theory of monads

• Freyd-Kelly, Categories of continuous functors, I

• Lawvere, Metric spaces, generalized logic and closed categories

• Kelly-Street, Review of the elements of 2-categories

• Street-Walters, Yoneda structures on 2-categories

• Johnstone, On a topological topos

• Kelly, Elementary observations on 2-categorical limits

• Adámek-Borceux-Lack-Rosický, A classification of accessible categories

• Lack, Codescent objects and coherence

• Shulman, Enriched indexed categories

Posted at 4:34 PM UTC | Permalink | Followups (5)

## October 6, 2013

### Unexpected Connections

#### Posted by Tom Leinster

On Wednesday I’ll give a half-hour talk to all the new maths PhD students in Scotland, called Unexpected connections. What should I put in it?

When British students arrive to do a PhD, they have already chosen a supervisor, and they have a fairly good idea of what their PhD topic will be. So, they’re in the mood to specialize. On the other hand, they’re obliged to take some courses, and here in Scotland the emphasis is on broadening — balancing that specialization by learning a wide range of subjects.

Some students don’t like this. They’re not undergraduates any more, they’ve decided what they want to work on, and they resent being made to study other things. My job is to enthuse them about the wider, wilder possibilities — to tell them about some of the amazing advances that have been made by bringing together parts of mathematics that might appear to be completely unconnected.

What are some compelling stories I could tell? What are your favourite examples of apparently disparate mathematical topics that have been brought together to extraordinary effect?

The more disconnected the topics seem to be, the better. Best of all would be stories that connect pure mathematics with either applied mathematics or statistics.

Posted at 11:08 PM UTC | Permalink | Followups (39)

## October 3, 2013

### Witten Looking Anew at the Jones Polynomial

#### Posted by David Corfield

Guest post by Bruce Bartlett

Today at the Clay Research Conference, Edward Witten gave a talk on A new look at the Jones polynomial of a knot. It was an opportune moment, 25 years after his original original paper.

Let me give a quick report-back. Hopefully the video and slides will be available on the Clay website at some point, but that may take some years!

Posted at 8:47 AM UTC | Permalink | Followups (5)

## October 2, 2013

### Who Ordered That?

#### Posted by Tom Leinster

Prize for the most peculiar theorem of the year must surely go to my colleague Natalia Iyudu and her collaborator Stanislav Shkarin, who recently proved the following conjecture of Kontsevich.

Start with a $3 \times 3$ matrix.

Take its transpose, then take the reciprocal of each entry, then take the inverse of the whole matrix.

Take the transpose of that, then take the reciprocal of each entry, then take the matrix inverse.

Take the transpose of that, then take the reciprocal of each entry, and then, finally, take the matrix inverse.

Theorem: Up to a bit of messing about, you’re back where you started.

What on earth does this mean? It’s not clear that anyone really knows.

Posted at 9:53 PM UTC | Permalink | Followups (11)

## September 24, 2013

### Clay Mathematics Conference

#### Posted by John Baez

A while back, Bruce Bartlett announced a Workshop on Quantum Mathematics and Computation at Oxford. But that’s part of a bigger thing:

Clay Research Conference, September 29 – October 4, 2013, Oxford University.

Posted at 1:34 AM UTC | Permalink | Followups (4)

### The Definition of Graph is Classified

#### Posted by Tom Leinster

Amid the (to me, highly disturbing) news that the state apparatuses of the US and UK have been secretly and systematically keeping records of our emails, our web browsing, our phone calls, our letters, our financial transactions and our physical location, here’s one half-smile’s-worth of light relief: the definition of graph is classified as “Top Secret”. (Update: or maybe not. See Tod’s comment below.)

(Click to see the news article this comes from, and to expand the tiny writing along the top and bottom which specifies the precise degree of top-secretness.)

This NSA slide uses the category theorists’ word object for a vertex of a graph. I’m relieved they don’t call the edges morphisms.

Posted at 12:52 AM UTC | Permalink | Followups (9)