The new Center for the Topology and Quantization of Moduli Spaces (CTQM) at Aarhus University wins my award for the most specialized pure mathematics institute. It will be hosting an opening symposium in a couple weeks featuring several talks that look interesting. If one is going to choose a specialized subject, this is an excellent one. Over the last couple decades the study of the moduli space of curves and of the moduli space of flat connections on a bundle over a surface has led to the discovery of many previously unsuspected relations between different parts of mathematics and between mathematics and physics. The fact that this subject is so fruitful remains somewhat of a mystery, and there is undoubtedly much more to be learned.
The National Academy of Sciences EPP 2010 committee should soon be producing a report with a 15 year plan for the future of high energy physics in the U.S. At the last meeting of the group last month Fermilab director Pier Oddone gave a presentation, focusing on opportunities in neutrino physics, strategy concerning the ILC, and the future of Fermilab.
Last week there was a conference on Particle Physics at the Verge of Discovery at Aspen. Lot of interesting talks on experimental particle physics. For an overview, see the summary talk by Paul Grannis.
The Templeton funded Foundational Questions Institute (FQXi) has announced that it will be publishing its inaugural request for proposals on Monday. This organization is led by Max Tegmark, who will be here at Columbia that day giving a physics department colloquium on From Derision Cosmology to Precision Cosmology. Unfortunately I have to be away that day and will miss the talk although I would have liked to attend it.
Steve Hsu, a physicist with a serious interest in economics, writes:
You might think science is a weighing machine, with experiments determining which theories survive and which ones perish. Healthy sciences certainly are weighing machines, and the imminence of weighing forces honesty in the voting. However, in particle physics the timescale over which voting is superseded by weighing has become decades — the length of a person’s entire scientific career. We will very likely (barring something amazing at the LHC, like the discovery of mini-black holes) have the first generation of string theorists retiring soon with absolutely no experimental tests of their *lifetime* of work. Nevertheless, some have been lavishly rewarded by the academic market for their contributions.
Scott Aaronson describes his field of computational complexity theory as “quantitative theology”, and goes on to note:
Incidentally, it’s ironic that some people derisively refer to string theory as “recreational mathematical theology.” String theory has to earn the status of mathematical theology — right now it’s merely physics! A good place for string theorists to start their theological training is this recent paper by Denef and Douglas.
Lee Smolin is giving a course on background independent quantum theories of gravity at the Perimeter Institute, with the lectures available online.
David Corfield has an interesting posting on research programs in mathematics. It includes links to various things from Ronald Brown including a new paper on Ehresmann’s work on groupoids, and his web-page on “Higher Dimensional Group Theory”. Among other things worth reading at Brown’s site is his account of the origins of Grothendieck’s “Pursuing Stacks”.
Corfield also points to an excellent list of problems in homotopy theory from Mark Hovey. Hovey starts off with the comment
The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory should go, analogous to the Weil conjectures in algebraic geometry or the Ravenel conjectures in our field in the late 70s. You can’t win the Fields Medal without a Fields Medal-winning problem; Deligne would not be DELIGNE without the Weil conjectures and Mike Hopkins would not be MIKE HOPKINS without the Ravenel conjectures.
I first met Mike Hopkins at a conference in Guanajuato around 1990, and he made a big impression on me. One thing that most impressed me (besides his joking comment that he went into topology because it was a field full of hard-drinking and living guys who got into gun-fights (this last part was a reference to Dennis Sullivan)), was the mathematical ambition he demonstrated. He said that he had up till then made his reputation proving other people’s conjectures, but now wanted to start making his own. Mike definitely followed through on this, since a sizable number of Hovey’s problems are inspired by him. His talk on elliptic cohomology in Guanajuato was a revelation, and he has over the years continued to work in that area, coming up with dramatic new ways of thinking about the subject.
Update: There is an extensive discussion of the Smolin lectures at Christine Dantas’s website.
Oh, Peter, thank you for the link to Brown’s account of Pursuing Stacks. I have never before come across this part of the story.
Hmm… I’m not sure that Hovey needs to worry that outsiders don’t care about algebraic topology. It seems pretty clear right now that a lot of future work in algebraic geometry is going to use techniques from algebraic topology. Especially in the theory of moduli spaces. Just look at the way Madsen, Weiss (& Tillman) proved Mumford’s conjecture. Or at Teleman & Woodward’s proof of the Newstead-Ramanan conjectures. Or have a look at Lurie’s thesis: He (and Toen & Vezzosi) are constructing an algebraic geometry where intersection theory always works; to do this, one must mix algebraic topology into the foundations of algebraic geometry.
Anyways, I don’t think the problem is that no one’s interested. The problem is that modern algebraic topology (model categories, spectra, etc,..) is difficult to learn. I personally found scheme theory
easier. The field is still waiting for a Hartshorne to come along and make everything accessible. (Peter May’s book is a nice start, but it’s doesn’t go nearly far enough.)
I think there is a real problem with the perception of homotopy theory (I just spent part of dinner arguing with a colleague about this), and Hovey undoubtedly has encountered it in is career. Mike Hopkins has done a great deal to broaden the scope of homotopy theory and to bring it back into contact with some of the deepest parts of the rest of mathematics. But many mathematicians are not really aware of this work, and one of the main reasons is the one you point out. It’s very difficult stuff to learn. A good expository text on the subject could do wonders.
Dear Peter Woit,
I have set a space in my blog to discuss Smolin´s lectures in detail, “The Hand of a Master Series”. Parts 1 and 2 of the lectures are already being discussed.
As I have put it, “Feel free to send your questions, answers, comments, doubts, criticisms, ideas, disscussions and feelings on these lectures“. Students and experts are all invited to
Thank you very much,
”…field full of hard-drinking and living guys who got into gun-fights (this last part was a reference to Dennis Sullivan)),…”
I’ve said it before and I’ll say it again, there are an awful lot of interesting tid-bits that are being alluded to but where the details are being witheld from us. Will someone fill in this gaping gap?
Mark Hovey writes in his page:
Concurrency theory. See the GETCO workshops. E.g.,
For the “hard-drinking and hard-living” details you’ll have to ask someone else, but the reference to Sullivan is to the fact that he was shot in the shoulder by gunmen trying to steal his car in Brazil (he had pulled over to the side of the road to take a nap). OK, I don’t think he was armed, so “gun-fight” is a bit of an exaggeration….
Thanks for mentioning the material about the Smolin lectures on your web-site, I’ll put in a link on the main page.
I see a great motivation for homotopy is that it is related to classification. So I have been led to be interested in the abstract structures underlying homotopy, in order to understand and calculate.
Also the aesthetic motive leads me to look for arguments which are easy and clear, because they follow a structure, and in order to make sure that I understand them. Not being as clever as many in the field, I need props and guidance, to explain to me, and I hope others also, why something is true, and also where it is going.
I can’t resist advertising here the new revised edition of my old topology book, to be available as `Topology and groupoids’, in print and e-version, in a few weeks – see my web site. This is the first step towards a full exposition of `Nonabelian algebraic topology’, a bigger job than expected!
It is amazing that even just double groupoids seem very complicated! They reflect I presume transitions of transitions. This should be useful in ………??
Dear Peter Woit,
I appreciate it, thank you!
Sir, I’m just an average guy with an infant-like understanding of all of these theories. I found your blog by searching wikipedia for String Theory after hearing someone mention it offhand. Wikipedia had a link to your blog as a critic of string theory. I read your article in American Scientist and I think I understand your criticism. Long story short, almost everything on your website is over my head, but I was wondering if you could provide me with some resources… book names or articles that might help me to begin to understand what you and your colleagues are discussing.
I am a scholar by no means and will understand if this is below you or a waste of your time. Thank you for any help.
I’ve written a book, coming out this fall, which is intended to be something like what you are looking for. There are lots of books out there, many of which are pretty good, you just have to realize that most of the discussion of string theory is way over-hyped. About the best book about particle physics that I know and that I recommend to people is “The Second Creation” by Crease and Mann. There was a discussion over at Cosmic Variance about popular books, see
The two books by Abraham Pais that Clifford Johnson recommends are quite good also.
“It is amazing that even just double groupoids seem very complicated! They reflect I presume transitions of transitions. This should be useful in ………??”
Some form of relational quantum mechanics! You said it!
The story sounds a bit similar to that of Gareth Evens, the Oxford philosopher of Language. He was shot in Brazil by a street kid, and then, tragically, died of the wound. Evans was quite young at the time.
Pity —Brazil has a good tradition of serious science and maths, I guess, most notably, Nachbin.
Thank you for your recommendation, I will have to see if I can find it at the library. I will look for your book when it comes out. Maybe after I read this book, I will stop by here and see if I can better understand what everyone is talking about. Thanks again.