The elementary construction of formal anafunctors

Some time ago I wrote notes constructing in a purely 2-categorical language a bicategory of anafunctors, starting from a 2-category $K$ equipped with a notion of cover, which in the original setting that Makkai studied, reduces to a trivial isofibration of categories. I always wanted to do something more with the construction, and I still do, but I thought it worthwhile to get the notes into a shape suitable for public consumption (at one point I had changed notational convention, and I found this week that the transition was half-way through a diagram!). So here they are:

The elementary construction of formal anafunctors, arXiv:1808.04552, doi:10.25909/5b6cfd1a73e55

Abstract: These notes give an elementary and formal 2-categorical construction of the bicategory of anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful.

As always, comments welcome.

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Choice quote from Venkatesh

ICM 2018 | "A lot of the time, when you do Math, you're stuck. But you feel privileged to work with it. You have a feeling of transcendence and feel like you've been part of something really meaningful." Akshay Venkatesh, 2018 Fields Medallist. #ICM2018 #ICM2018Rio

— ICM2018 (@ICM_2018) August 1, 2018

Gowers on Scholastica

Timothy Gowers, as many would know is a Fields medallist and has become somewhat of a spokesperson, or at the least a figurehead, for open access issues in mathematics publishing. He spearheaded the Cost of Knowledge boycott of Elsevier, helped found the open access journals Forum of Mathematics: Pi and Sigma and more recently, Discrete Analysis and the new Advances in Combinatorics. The difference between the former two and the latter is that the FoM journals are published by a commercial publisher and have non-zero article processing charges (APCs), typically paid for out of research funds, or library OA funds etc, whereas DA and AinC is an ‘arXiv-overlay’ journal: the final versions of articles are stored on the arXiv, and the cost to the journal for each article is O($10), which is covered by a grant/donated funding. Needless to say, these journals are part of the Free Journal Network, the aim of which is to be a loose confederation of open access journals that are free to publish in and meet modern standards of ‘openness’. The journal websites for DA and AinC are hosted by Scholastica, and Tim recently did an interesting interview with Scholastica about the whole business.

 

 

 

What is the curl of a vector field, really?

The impetus for this post came from David Butler (blog, Twitter, also found on #mtbos, the “math-twitter-blog-o-sphere”), who wanted an explanation of what the curl of a vector field is without recourse to explanations involving physical phenomena, like the electromagnetic field, or fluid flow and little rotating paddles and so on. What he wanted was a description of how the curl is a derivative, that is, the linearisation of something, much like the gradient of a vector field (or indeed an ordinary derivative) tells us about the linearisation of a real-valued function. This led me on an interesting journey in how to turn how I think of the curl (as the 2-form that is the exterior derivative of a 1-form), into something palatable for the students learning it for the first time, without implicitly teaching them new material in order to grasp the explanation. In the end, it didn’t quite turn out like I expected, and the answer was kind-of hidden in existing theory of continuum mechanics; I grew to suspect that what I was trying to do had been done before, but didn’t know where to look. Click through to get what I feel is a comprehensive answer, complete with pretty pictures!

Nalini Joshi nominated for VP of the IMU!

In two weeks, I’ll be flying to @ICM_2018 as Australia’s first ever nominee for Vice-President of the International Mathematics Union.

— Nalini Joshi (@monsoon0) July 12, 2018

 

Edit (31 July): And she is in! Together with Loyiso Nongxa of South Africa, Nalini will be Vice-President for 2019-2022 (The IMU has a tradition of having more than one VP, in case that looks odd). The President will be Carlos Kenig.

Elsevier finally pull the pin on Germany

Elsevier last week stopped thousands of scientists in Germany from reading its recent journal articles, as a row escalates over the cost of a nationwide open-access agreement.

The move comes just two weeks after researchers in Sweden lost access to the most recent Elsevier research papers, when negotiations on its contract broke down over the same issue.

Dutch publishing giant cuts off researchers in Germany and Sweden, Nature News, 19 July 2018, doi:10.1038/d41586-018-05754-1

Special issue of Geombinatorics on recent progress on the chromatic number of the plane

The recent breakthrough work by de Grey that showed the chromatic number of the plane could not be equal to 4 (and so must be 5, 6 or 7) has been published, along with a few other papers in a special issue of the journal Geombinatorics. There are free copies of all the articles in this subscription journal around the place, so I thought I’d gather links to all of them here.

• Breakthrough in My Favorite Open Problem of Mathematics: Chromatic Number of the Plane, Alexander Soifer (this is a shorter version of the full published article, which has a slightly different title: Progress in My Favorite Open Problem of Mathematics, Chromatic Number of the Plane: An Étude in Five Movements)

• The chromatic number of the plane is at least 5, Aubrey D.N.J. de Grey (arXiv:1804.02385)

• Computing Small Unit-Distance Graphs with Chromatic Number 5, Marijn J.H. Heule (arXiv:1805.12181)

• The Hadwiger-Nelson problem with two forbidden distances, Geoffrey Exoo, Dan Ismailescu (arXiv:1805.06055)

Apparently, Exoo and Ismailescu managed to rule out the case that the chromatic number is 4 independently and at about the same time as de Grey, but wanted to improve the construction and shrink the graph they used, and so were scooped while they kept working in secret.