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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.
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
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 ‘-mathematics’.
As before, you can read John Simanyi’s wonderful notes in LaTeX. If you find mistakes, please let me know.
October 25, 2016
The Kan Extension Seminar Returns
Posted by Emily Riehl
In early 2014, the -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.
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).
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!
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”.
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.