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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 (13)

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)

Linear Algebraic Groups (Part 3)

Posted by John Baez

This time we touch on some other aspects of algebraic group theory, again using the example of projective geometry. We describe the decomposition of projective space into ‘Bruhat cells’. These let us count the points of projective spaces over finite fields, which gets us a wee bit deeper into the fascinating and somewhat mysterious topic of ‘qq-mathematics’.

As before, you can read John Simanyi’s wonderful notes in LaTeX. If you find mistakes, please let me know.

Posted at 1:17 AM UTC | Permalink | Post a Comment

October 25, 2016

The Kan Extension Seminar Returns

Posted by Emily Riehl

In early 2014, the nn-Category Café hosted the Kan Extension Seminar, a graduate reading course in category theory modeled after Daniel Kan’s eponymous reading course in algebraic topology at MIT. My experience with the seminar, described here, was overwhelming positive, so I am delighted to announce that we’re back. Alexander Campbell, Brendan Fong, and I are organizing “Kan II” in early 2017 and we are currently soliciting applications for seminar participants.

Posted at 5:37 PM UTC | Permalink | Followups (6)

October 20, 2016

Linear Algebraic Groups (Part 2)

Posted by John Baez

This time we show how projective geometry ‘subsumes’ Euclidean, elliptic and hyperbolic geometry. It does so in two ways: the projective plane includes all 3 other planes, and its symmetry group contains their symmetry groups.

By the time we understand this, we’re almost ready to think about geometry as a subject that depends on a choice of group. But we’re also getting ready to think about algebraic geometry (for example, projective varieties).

Posted at 1:07 AM UTC | Permalink | Post a Comment

October 17, 2016

Linear Algebraic Groups (Part 1)

Posted by John Baez

I’m teaching an elementary course on linear algebraic groups. The main aim is not to prove a lot of theorems, but rather to give some sense of the main examples and the overall point of the subject. I’ll start with the ideas of Klein geometry, and their origin in old questions going back almost to Euclid.

John Simanyi has been taking wonderful notes in LaTeX, so you can read those!

Posted at 12:52 AM UTC | Permalink | Followups (7)

October 10, 2016

Jobs at Edinburgh

Posted by Tom Leinster

I’m pleased to announce that we’re advertising two Lectureships in “algebra, geometry & topology and related fields such as category theory and mathematical physics”. Come and join us! We’re a happy, well-resourced department with a very positive atmosphere. The algebra/geometry/topology group provides an excellent home for a category theorist.

To be clear, these positions are for practical purposes permanent, i.e. as close as the UK gets to tenure. There’s no one-to-one correspondence between UK and US job titles, so I’ll just say that Lecturer is the usual starting position for someone in their first permanent job, followed by Senior Lecturer, Reader, then Professor. The ad adds “Exceptionally, the appointments may be to Readership”.

Posted at 1:51 PM UTC | Permalink | Post a Comment

October 4, 2016

Mathematics Research Community in HoTT

Posted by Emily Riehl

I am delighted to announce that from June 4-10, 2017, there will be a workshop on homotopy type theory as one of the AMS’s Mathematical Research Communities.

The MRC program, whose workshops are held in the “breathtaking mountain setting” of Snowbird Resort in Utah,

nurtures early-career mathematicians — those who are close to finishing their doctorates or have recently finished — and provides them with opportunities to build social and collaborative networks to inspire and sustain each other in their work.

The organizers for the HoTT MRC include our fearless leader Chris Kapulkin, Dan Christensen, Dan Licata, Mike Shulman and myself.

Posted at 1:47 PM UTC | Permalink | Post a Comment

September 26, 2016

Euclidean, Hyperbolic and Elliptic Geometry

Posted by John Baez

There are two famous kinds of non-Euclidean geometry: hyperbolic geometry and elliptic geometry (which almost deserves to be called ‘spherical’ geometry, but not quite because we identify antipodal points on the sphere).

In fact, these two kinds of geometry, together with Euclidean geometry, fit into a unified framework with a parameter ss \in \mathbb{R} that tells you the curvature of space:

  • when s>0s \gt 0 you’re doing elliptic geometry

  • when s=0s = 0 you’re doing Euclidean geometry

  • when s<0s \lt 0 you’re doing hyperbolic geometry.

This is all well-known, but I’m trying to explain it in a course I’m teaching, and there’s something that’s bugging me.

It concerns the precise way in which elliptic and hyperbolic geometry reduce to Euclidean geometry as s0s \to 0. I know this is a problem of deformation theory involving a group contraction, indeed I know all sorts of fancy junk, but my problem is fairly basic and this junk isn’t helping.

Posted at 2:25 AM UTC | Permalink | Followups (24)