This fall, the topology group at OSU is reading through Dani Wise’s lecture notes on cube complexes, based on his series of talks at the CBMS-NSF conference back in 2011. Henry Wilton and Daniel Moskovich have written on this blog about Wise’s work and its role in the proof of the Virtual Haken Conjecture. This is just a quick note to say how impressed I’ve been with the lecture notes. They start from the very beginning, include a lot of good examples and have proved to be very accessible for all of us non-experts (which includes me).
It’s just too bad that Wise’s notes are no longer available on the conference web page. (Now that a paper copy is available from the AMS, the PDF file has been replaced with a note saying that the editor insisted they be taken down.) You can still e-mail Dani Wise to request a copy, but I expect that some people (such as beginning graduate students) might be reluctant to e-mail someone they don’t know like this. I can assure you, he was very gracious when I asked him for a copy and seems to be very eager to distribute the notes widely. But, if you have any thoughts on how the PDF file could be distributed more efficiently, I would love to hear about it in the comments.
Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology. Almost everything we blog about here has the imprint of his amazing mathematics. Bill was always very generous with his ideas, and his presence in the community will be horribly missed. Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:
Cubulation and the proof of the VHC/VFC have been featured prominently on this blog, and quite rightly so. It constitutes a major advance in low dimensional topology, and the proof itself is such a happy story. Thurston’s prophetic unexpected 1982 conjectures of how nice the fundamental group of a hyperbolic 3-manifold must be all proved to be true (such a group turns out to be LERF, virtually fibred, virtually biorderable, linear over the integers, etc.). And it happened in the nicest way possible, without huge mathematical machines (as was the case for e.g. Hironaka’s resolution of singularities), without huge computer assistance (as was the case for e.g. the Four Colour Theorem), without controversity over correctness of the proof (compare e.g. Wiles’s first proof of Fermat’s Last Theorem), with a great deal of conceptual understanding gained (contrast again with the Four Colour Theorem), without hard-to-read outlined arguments which require non-trivial unpacking (contrast with Perelman’s proof of Geometrization), without being unreadably long (contrast with classification of finite simple groups), with essential new ideas and new tools developed, together with a theorem which might be more valuable than the conjecture itself (the Virtually Compact Special Theorem- compare with Geometrization being more valuable than the Poincaré Conjecture).
Indeed, for me this might be the happiest end ever for a quest to prove a mathematical conjecture. What more could one wish for?
I just got back from a trip to North America, where I visited Toronto, Kansas, and Montreal. I have a lot to say about my visits to all 3 places really, but today I want to talk about my visit to The Cubulator Dani Wise, and some snippets of what he taught me, and some of my vague mathematically unmeaningful thoughts about some key ideas of the cubulation story (for a more professional account of related things, read Danny Calegari’s coverage). (more…)
Anyone who has been following this blog knows that the last few months and years have seen a bewildering amount of progress in what we know about 3-manifolds, and particularly their fundamental groups. Matthias Aschenbrenner, Stefan Friedl and I have recently posted a survey paper on the arXiv, the aim of which is to summarise these recent developments and state, as definitively as possible, what we currently know about 3-manifold groups.
It’s a dauntingly large subject, and there are inevitably many errors and omissions. No doubt we will need to produce an updated version soon. For this reason, we actively encourage comments. We’re particularly keen to hear about mistakes or misattributions, but even if we’ve just omitted your favourite paper (even if you wrote it yourself!) please do drop us a line and let us know. We’d love to hear from you.
Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.
The news of Ian Agols claimed proof of the Virtually Haken/fibered conjectures is very exciting and I look forward to reading Danny Calegari’s summaries of Ian’s talks. But at the same time, I remember the time after Perelmann announced his proof of the Poincare conjecture and the dire predictions of the end of 3-manifold topology, and I wonder what the aftermath will be this time.
The most visible aspect of this subject has always been its open problems – The Poincare conjecture at first, then Thurston’s suite of problems, including the VHC. Many (most? all?) fields are guided by major open problems, but 3-dimensional topology stands out for borrowing most of its most powerful techniques from outside – algebra, geometry, dynamical systems, gauge theory, etc. These connections have invigorated and popularized the field, but they have also created a situation in which when a major problem falls, so does the field’s public image.
Over at Geometry and the Imagination, Danny Calegari is reporting live from Paris on talks by Agol and Manning on the announced proof of the VHC: Part I, Part II, Part III.
This just in: Ian Agol (UC Berkeley), speaking at the Institut Henri Poincaré today, has announced a proof of the very same Wise’s Conjecture that I blogged about just last week! In particular, this implies the Virtually Haken Conjecture. His proof is based on joint work with Daniel Groves (UI Chicago) and Jason Manning (SUNY Buffalo). It makes heavy use of the work of Dani Wise (McGill) on the Virtually Fibred Conjecture, as well as the proof of the Surface Subgroup Conjecture by Jeremy Kahn (Brown) and Vlad Markovic (Caltech).
I’ve mentioned here several times the work of Wise on residual properties of certain word-hyperbolic groups, specifically those of Haken hyperbolic 3-manifolds. You can now view all 10 of Dani’s talks at the NSF-CBMS conference (as well as all the other talks) at the conference webpage. The picture and audio quality is quite reasonable considering the setup that was used, and they are certainly watchable.
I really wish more conferences did this. While it’s certainly true that the benefits of attending a conference go far beyond the content of the talks themselves, I think it’s still quite valuable to have this online for those who weren’t able to addend.
This blog has mentioned several times Dani Wise’s work on subgroup separability properties for certain word-hyperbolic groups [1, 2, 3]. In August, there will be a CBMS-NSF conference at CUNY focusing on this work, and the reason for this post is that one major part of Dani’s work is now available on the conference website.
(Thanks to Jason Behrstock and Ian Agol for telling me that the preprint had been posted there.)