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Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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 (56)

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 (20)

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)

December 15, 2016

Field Notes on the Behaviour of a Large Assemblage of Ecologists

Posted by Tom Leinster

I’ve just come back from the annual conference of the British Ecological Society in Liverpool. For several years I’ve had a side-interest in ecology, but I’d never spent time with a really large group of ecologists before, and it taught me some things. Here goes:

Posted at 12:44 AM UTC | Permalink | Followups (21)

November 29, 2016

Quarter-Turns

Posted by Tom Leinster

Teaching linear algebra this semester has made me face up to the fact that for a linear operator TT on a real inner product space, Tx,x=0xT *=T \langle T x, x \rangle = 0 \,\, \forall x \,\, \iff \,\, T^\ast = -T whereas for an operator on a complex inner product space, Tx,x=0xT=0. \langle T x, x \rangle = 0 \,\, \forall x \,\, \iff \,\, T = 0. In other words, call an operator TT a quarter-turn if Tx,x=0\langle T x, x \rangle = 0 for all xx. Then the real quarter-turns correspond to the skew symmetric matrices — but apart from the zero operator, there are no complex quarter turns at all.

Where in my mental landscape should I place these facts?

Posted at 11:14 PM UTC | Permalink | Followups (24)

Linear Algebraic Groups (Part 8)

Posted by John Baez

The course proceeds apace, but my notifications here have slowed down as I become over-saturated with work.

In Part 8, I began explaining a bit of algebraic geometry. Following the general pattern of this course I took a quasi-historical approach, explaining some older ideas before moving on to newer ones. I’m afraid I never got to explaining schemes. That’s a tragedy, but hey—life is full of tragedies, nobody will notice this one. Affine schemes is all I had time for, despite the fact that I was discussing a lot of projective geometry. And before explaining affine schemes, it seemed wise to mention some earlier ideas and their defects.

Posted at 1:36 AM UTC | Permalink | Followups (2)

November 16, 2016

Category Theory in Context

Posted by Emily Riehl

In my final year at Harvard and again in my first year at Johns Hopkins, I had an opportunity to teach an advanced undergraduate/beginning graduate-level topics course entitled “Category Theory in Context.” Its aim was to provide a first introduction to the basic concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and Kan extensions — while simultaneously discussing the implications of these ideas in a wide variety of areas of mathematics on which category theory sheds light.

I thought teaching this course would provide a fun opportunity to collect as many examples of this kind as I could, for which I solicited widely — more about this below. This provided the impetus to write lecture notes. And now they have been published by Dover Publications in their new Aurora: Modern Math Originals series.

graphs

I extremely grateful to Dover for granting me permission to host a free PDF copy of the book on my website. This version is in some sense even better than the published version, in that I have been able to correct a handful of typos that were discovered after the print version was already in press.

Posted at 10:28 PM UTC | Permalink | Followups (9)

November 10, 2016

Linear Algebraic Groups (Part 7)

Posted by John Baez

One of the less obvious but truly fundamental realizations in group theory is the importance of the ‘parabolic subgroups’ of a linear algebraic group. Today we’ll sneak up on this realization using the example of GL(n)\mathrm{GL}(n).

We’ve already seen the Klein geometry corresponding to this group has important kinds of figures — points, lines, planes, etc. — whose stabilizers are certain nice groups called ‘maximal parabolic subgroups’ of GL(n)\mathrm{GL}(n). But there are also important figures build from these, like ‘a point lying on a line’, or ‘a line lying on a plane’. These are called ‘flags’, and their stabilizers are called ‘parabolic subgroups’. Today we’ll work out what these parabolic subgroups of GL(n)\mathrm{GL}(n) are like. Especially important is the smallest one, called the ‘Borel’.

With this intuition in hand, we’ll want to generalize all these concepts to an arbitrary linear algebraic group. Amazingly, you can just hand someone such a group, and they can figure out the important kinds of geometrical figures in its Klein geometry, by determining its parabolic subgroups!

Posted at 2:13 AM UTC | Permalink | Followups (3)

November 7, 2016

Linear Algebraic Groups (Part 6)

Posted by John Baez

When you’re doing math, if you ever want to keep things from getting too wispy and ethereal, it’s always good to count something. In fact, even if counting were good for nothing else — a strange counterfactual, that — mathematicians might have invented it for this purpose. It’s a great way to meditate on whatever structures one happens to be studying. It’s not the specific numbers that matter so much, it’s the patterns you find.

Last time we introduced Grassmannians as a key example of Klein’s approach to geometry, where each type of geometrical figure corresponds to a homogeneous space. Now let’s count the number of points in a Grassmannian over a finite field. This leads to a qq-deformed version of Pascal’s triangle. Then, if we categorify the recurrence relation defining the qq-binomial coefficients, we’ll understand the Bruhat cells for Grassmannians over arbitrary fields!

Posted at 1:00 AM UTC | Permalink | Followups (3)

November 3, 2016

Linear Algebraic Groups (Part 5)

Posted by John Baez

Now let’s look at projective geometry from a Kleinian viewpoint. We’ll take the most obvious types of figures — points, lines, planes, and so on — and see which subgroups of GL(n)\mathrm{GL}(n) they correspond to. This leads us to the concept of ‘maximal parabolic subgroup’, which we’ll later generalize to other linear algebraic groups.

We’ll also get ready to count points in Grassmannians over finite fields. For that, we need the qq-deformed version of binomial coefficients.

Posted at 1:35 AM UTC | Permalink | Post a Comment

October 30, 2016

Linear Algebraic Groups (Part 4)

Posted by John Baez

This time I explain some axioms for an ‘abstract projective plane’, and the extra axiom required to ensure an abstract projective plane comes from a field. Yet again the old Greek mathematicians seem to have been strangely prescient, because this extra axiom was discovered by Pappus of Alexandria sometime around 340 AD! For him it was a theorem in Euclidean geometry, but later it was realized that a cleaner statement involves only projective geometry… and later still, it was seen to be a useful axiom.

For details, read the notes.

Posted at 1:25 AM UTC | Permalink | Followups (1)

October 26, 2016

Higher Structures Journal

Posted by John Baez

Michael Batanin, Ralph Kaufmann and Martin Markl are the editors of a new diamond open access journal called Higher Structures. The managing editor is Mark Weber, and here’s the editorial board:

Clemens Berger, Université Nice-Sophia Antipolis

Vladimir Dotsenko, Trinity College Dublin, the University of Dublin

Tobias Dyckerhoff, Hausdorff Center for Mathematics

Benoit Fresse, Université de Lille

Richard Garner, Macquarie University

André Henriques, Universiteit Utrecht

Joachim Kock, Universitat Autònoma de Barcelona

Stephen Lack, Macquarie University

Andrey Lazarev, Lancaster University

Muriel Livernet, Université Paris Diderot

Michael Makkai, McGill University

Yuri Manin, Max Planck Institute for Mathematics

Ieke Moerdijk, Universiteit Utrecht

Amnon Neeman, Australian National University

Maria Ofelia Ronco, Universidad de Talca

Jiří Rosický, Masaryk University

James Stasheff, University of Pennsylvania

Ross Street, Macquarie University

Bertrand Toën, Université de Toulouse

Boris Tsygan, Northwestern University

Bruno Vallette, Université Paris 13

Michel Van den Bergh, Universiteit Hasselt

Alexander Voronov, University of Minnesota

Posted at 4:08 PM UTC | Permalink | Followups (4)