What makes the Monster Lie Algebra special? May 26, 2014Posted by Scott Carnahan in group theory, mathematical physics, representation theory.
add a comment
This is a post I’d been meaning to write for several years, but I was finally prompted to action after talking to some confused physicists. The Monster Lie Algebra, as a Lie algebra, has very little structure – it (or rather, its positive subalgebra) is quite close to being free on countably infinitely many generators. In addition to its Lie algebra structure, it has a faithful action of the monster simple group by Lie algebra automorphisms. However, the bare fact that the monster acts faithfully on the Lie algebra by diagram automorphisms is not very interesting: the almost-freeness means that the diagram automorphism group is more or less the direct product of a sequence of general linear groups of unbounded rank, and the monster embeds in any such group very easily.
The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.
The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.
Elsevier in Australia May 21, 2014Posted by Scott Morrison in elsevier, evil journals, publishing.
I’ve just got back from talking to Roxanne Missingham, the University Librarian here at ANU, about Elsevier, and I want to quickly report on what I learnt.
I don’t yet have any of the juicy numbers revealing what libraries are paying for their Elsevier subscriptions (as Timothy Gowers has been doing in the UK; if you haven’t read his post do that first!). Nevertheless there are some interesting details.
Essentially all the Australian universities, excepting a few tiny private institutes, subscribe to the Freedom collection (this is the same bundle that nearly everyone is forced into subscribing to). The contracts are negotiated by CAUL (the Council of Australian University Librarians).
My librarian was very frank about Article Processing Charges (APCs) constituting double-dipping, whatever it is that Elsevier and the other publishers say. The pricing of journal bundles is so opaque, and to the extent we understand it primarily based on the historical contingencies of print subscription levels more than a decade ago, that in practice the fraction of articles in a subscription bundle for which APCs have been paid has no meaningful effect on the prices libraries pay for their bundles.
I think this point needs wider dissemination amongst mathematicians — whatever our complaints about APCs inhibiting access to journals for mathematicians without substantial funding, we are just plain and simple being ripped off. Gold open access hybrid journals are a scam.
Now, on to some details about contracts. First, my librarian confirmed the impression from Gowers’ investigations in the UK — bundle pricing is based largely on historical spending on print subscriptions, with annual price increases. Adding some interesting context on the numbers we’re now seeing out of the UK, she told me that the UK is widely perceived as having received a (relatively) great deal from Elsevier, in terms of annual price increases. If the UK numbers scared you, be aware that here in Australia we may well have it worse. A curious anecdote about historical pricing of subscriptions is that one division of CSIRO happened to have cancelled most of their print journals the year before they took out an electronic subscription with a commercial publisher, and as a result got an excellent deal. The Australian universities have apparently mostly signed confidentiality agreements regarding their journal subscription costs (as we expect, by now), but my understanding of the conversation was that the ANU in particular had not.
Finally, my librarian pointed out that doing what I hope to do next, namely use the FOI act to obtain detailed information on Elsevier subscription costs, may be counterproductive, as the most likely result of unusual discrepancies in pricing being revealed is some libraries simply having budgets cut, rather than actually giving the negotiators any more power in the future. I got the impression she’d talked to other Australian librarians about this, and there was some amount of nervousness.
I’ve been told I should go talk to Andrew Wells, the librarian at UNSW, and after posting this I’m going to get in touch with him!
The many principles of conservation of number March 4, 2014Posted by David Speyer in Uncategorized.
In algebraic geometry, we like to make statements like: “two conics meet at points”, “a degree four plane curve has bitangents”, “given four lines in three space, there are lines that meet all of them”. In each of these, we are saying that, as some parameter (the conics, the degree four curve, the lines) changes, the number of solutions to some equation stays constant. The “principle of conservation of number” refers to various theorems which make this precise.
In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely because there are many different ways to make these details precise. I decided to try and write down all the basic results I could think of along these lines.
Mathematical Research Community on Cluster Algebras in Utah this summer February 26, 2014Posted by David Speyer in Uncategorized.
This June 8 to 14, there will be a week long gathering in Snowbird, Utah for young mathematicians working on cluster algebras. The target audience here are either current graduate students, or people with Ph. D. in the last 3 or so years, who would be ready to start working on problems in cluster algebras. The hope is to spend a lot of time getting collaborations and projects going during the week. The organizers are Michael Gekhtman, Mark Gross, Gregg Musiker, Gordana Todorov and me.
We still have room for a number more applicants, so we would like to encourage more of you to apply. Please note that the application deadline of March 1 is firm.
Australian Research Council journal list February 24, 2014Posted by Scott Morrison in Uncategorized.
(This post may only be of interest to Australian mathematicians; sorry!)
Summary: A number of mathematics journals (e.g. Quantum Topology, Forum of Mathematics Sigma and Pi, and probably many others), are not listed on the new official journal list in Australia. Please, help identify missing journals, and submit feedback via http://jacci.arc.gov.au/.
Every few years the Australian Research Council updates their “official list of journals”. One might wonder why it’s necessary to have such a list, but nevertheless it is there, and it is important that it is accurate because the research outputs of Australian mathematicians are essentially filtered by this list for various purposes.
There is a new draft list out, and the purpose of this post is to coordinate finding missing journals, and to ensure that interested mathematicians submit feedback before the deadline of March 15. Please note that while in the past this list included dubious rankings of journals, the current list is just meant to track all peer reviewed journals in each subject. Having a journal missing entirely means that some published papers will not be counted in measures of a department’s or university’s research output.
You can access the full list here, just journals marked as mathematics here, and just the journals marked a pure mathematics here. These are not the “official” lists, which you have to create an account (follow the instructions at http://www.arc.gov.au/era/current_consult.htm) to view, and even then only an Excel version is available. I hope that by making these mathematics specific lists available in a standard format, more mathematicians will take the time to look over the list.
Please look through the lists. If you see something missing, please comment here so we all know about it. In any case, please submit feedback via http://jacci.arc.gov.au/ (you’ll have to create an account first) recommending inclusion of the journals identified so far. Submitting a missing journal requires identifying an article published in it by an Australia author; feel free to add this information here as well if appropriate. (Thanks to Anthony Henderson for pointing out this detail!)
It is also possible to submit additional “FoR” (field of research) codes for journals on the list, and this may be of interest to people publishing cross-disciplinary research. Feel free to make suggestions along these line here too: the AustMS has been advised that “multiple responses, rather than a single AustMS one, will carry more weight on this aspect”.
Course on categorical actions February 7, 2014Posted by Ben Webster in Shamelss Self Promotion.
I have the excellent luck to be sending this semester in Paris, thanks to the Fondation Sciences Mathématiques de Paris. Part of the deal is that I’m giving a weekly course at the “graduate level” (though I think I have more professors than graduate students in the course) on higher representation theory. Also thanks to FSMP, the course is being videotaped and posted online; the first installment is up here. I’m also posting the videos and additional commentary on a WordPress site; if you have any questions, you can always ask them there (or here, but maybe it’s more germane there).
Postdocs at ANU January 23, 2014Posted by Scott Morrison in jobs.
We intend that these will be 2 year positions, with minimal teaching requirements.
There is an informal description of the jobs at http://tqft.net/web/postdoc, including some information about the grants funding these positions. The official ad is online at http://jobs.anu.edu.au/PositionDetail.aspx?p=3736, and you can find it on MathJobs at http://www.mathjobs.org/jobs/jobs/5678.
Please contact us if you have questions, and please encourage good Ph.D. students (especially with interests in subfactors, fusion categories, categorification, or related subjects) to apply!
Mathematics Literature Project progress January 6, 2014Posted by Scott Morrison in Uncategorized.
We’ve made some good progress over at the Mathematics Literature Project. In particular, we’ve completely analyzed the 2013 issues of five journals:
(The colour coded bars show the fractions of papers available on the arXiv, available on authors’ webpages, and not freely accessible at all; these now appear all over the wiki, but unfortunately don’t update automatically. Over at the wiki you can hover over these bars to get the numerical totals, too.)
Thanks everyone for your contributions so far! If you’ve just arrived, check out the tutorial I made on editing the wiki. Now, it’s time to do a little planning.
What questions should we be asking?
Here’s one we can start to answer right away.
What fraction of recent papers are available on the arXiv or on authors webpages?
For good generalist journals (e.g. Adv. Math. and Annals), almost everything! For subject area journals, there is wide variation (probably mostly depending on traditions in subfields): AGT is almost completely freely accessible, while Discrete Math. is at most half.
I hope we’ll soon be able to say this for many other journals, too.
Here’s the question I really want to have answers for:
Does being freely accessible correlate well with quality?
It’s certainly tempting to think so, seeing how accessible Advances and Annals are. I think to really answer this question we’re going to have to classify all the articles in slightly older issues (2010?) and then start looking at the citation counts for articles in the two pools. If we get coverage of more journals, we can also look for the correlation between, say, impact factor and the ratio of freely accessible content.
I don’t want to just list every journal on the wiki; it’s best if editors (and the helpful bots working in the background) can focus attention and enjoy the pleasures of finishing off issues and journals. Suggestions for journals to add next welcome in the comments. I’ve already included the tables of contents for the Journal of Number Theory, and the Journal of Functional Analysis. (It will be nice to be able to make comparisons between JFA and GAFA, I think.)
I’ve been working with some people on automating the entry of data in the wiki (mainly by using arXiv metadata; there are actually way more articles there with journal references and DOIs than I’d expected). Hopefully this will make the wiki editing experience more fun, as a lot of the work will have already been done, and humans just get to handle the hard and interesting cases.
Tags: mathematical literature
It would be nice to know how much of the mathematical literature is freely accessible. Here by ‘freely accessible’ I mean “there is a URL which, in any browser anywhere in the world, resolves to the contents of the article”. (And my intention throughout is that this article is legitimately hosted, either on the arxiv, on an institutional repository, or on an author’s webpage, but I don’t care how the article is actually licensed.) I think it’s going to be okay to not worry too much about discrepancies between the published version and a freely accessible version — we’re all grown ups and understand that these things happen. Perhaps a short comment field, containing for example “minor differences from the published version” could be provided when necessary.
This post outlines an idea to achieve this, via a human editable database containing the tables of contents of journals, and links, where available, to a freely accessible copy of the articles.
It’s important to realize that the goal is *not* to laboriously create a bad search engine. Google Scholar already does a very good job of identifying freely accessible copies of particular mathematics articles. The goal is to be able to definitively answer questions such as “which journals are primarily, or even entirely, freely accessible?”, to track progress towards making the mathematical literature more accessible, and finally to draw attention to, and focus enthusiasm for, such progress.
I think it’s essential, although this is not obvious, that at first the database is primarily created “by hand”. Certainly there is scope for computer programs to help a lot! (For example, by populating tables of contents, or querying google scholar or other sources to find freely accessible versions.) Nevertheless curation at the per-article level will certainly be necessary, and so whichever route one takes it must be possible for humans to edit the database. I think that starting off with the goal of primarily human contributions achieved two purposes: one, it provides an immediate means to recruit and organize interested participants, and two, hopefully it allows much more flexibility in the design and organization of the collected data — hopefully many eyes will reveal bad decisions early, while they’re easy to fix.
That said, we better remember that eventually computers may be very helpful, and avoid design decisions that make computer interaction with the database difficult.
What should this database look like? I’m imagining a website containing a list of journals (at first perhaps just one), and for each journal a list of issues, and for each issue a table of contents.
The table of contents might be very simple, having as few as four columns: the title, the authors, the link to the publishers webpage, and a freely accessible link, if known. All these lists and table of contents entries must be editable by a user — if, for example no freely accessible link is known, this fact should be displayed along with a prominent link or button which allows a reader to contribute one.
At this point I think it’s time to consider what software might drive this website. One option is to build something specifically tailored to the purpose. Another is to use an essentially off-the-shelf wiki, for example tiddlywiki as Tim Gowers used when analyzing an issue of Discrete Math.
Custom software is of course great, but it takes programming experience and resources. (That said, perhaps not much — I’m confident I could make something usable myself, and I know people who could do it in a more reasonable timespan!) I want to essentially ignore this possibility, and instead use mediawiki (the wiki software driving wikipedia) to build a very simple database that is readable and editable by both humans and computers. If you’re impatient, jump to http://tqft.net/mlp and start editing! I’ve previously used it to develop the Knot Atlas at http://katlas.org/ with Dror Bar-Natan (and subsequently many wiki editors). There we solved a very similar set of problems, achieving human readable and editable pages, with “under the hood” a very simple database maintained directly in the wiki.
From the drawers of the museum December 12, 2013Posted by Noah Snyder in fusion categories, quantum groups, subfactors, Uncategorized.
One of my amateur interests is paleontology. Paleontologists looking for new examples have two options: go out in the field and dig up a new example, or go looking through drawers of museums to find old examples that had been overlooked. In this blog post I want to give an interesting example of the latter kind of research being useful in mathematics. Namely in discussions with Zhengwei Liu, we realized that an old example of Ocneanu’s gives an answer to a question that was thought to be open.
One of the central problems in fusion categories is to determine to what extent fusion categories can be classified in terms of finite groups and quantum groups (perhaps combined in strange ways) or whether there are exceptional fusion categories which cannot be so classified. My money is on the latter, and in particular I think extended Haagerup gives an exotic fusion category. However, there are a number of examples which seem to involve finite groups, but where we don’t know how to classify them in terms of group theoretic data. For example, the Haagerup fusion category has a 3-fold symmetry and may be built from or (as suggested by Evans-Gannon). The simplest examples of these kind of “close to group” categories, are called “near-group categories” which have only one non-invertible object and have the fusion rules
for some group of invertible objects . A result of Evans-Gannon (independently proved by Izumi in slightly more generality), says that outside of a reasonably well understood case (where and the category is described by group theoretic data), we have that must be a multiple of . There are the Tambara-Yamagami categories where , and many examples (E6, examples of Izumi, many examples of Evans-Gannon) where
Here’s the question: Are there examples where n is larger than ?
It turns out the answer is yes! In fact the answer is given by the -graded part of the quantum subgroup of quantum from Ocneanu’s tables here. I’ll explain why below.