## Elsevier Gives Up On Research Work Act

27 February, 2012

A small victory for the rebel forces:

Yay! Let’s keep up the pressure, crush the Research Works Act, and move to take the offensive!

In case you haven’t heard yet: this nasty bill would stop the National Institute of Health from making taxpayer-funded research freely available to US taxpayers. It’s supported by the Association of American Publishers (AAP)—but various AAP members, including MIT Press, Rockefeller University Press, Nature Publishing Group, and the American Association for the Advancement of Science have already come out against it. Now, put under pressure by the spreading boycott, Elsevier has dropped its support.

However, they make it crystal clear that this is just a tactical retreat:

… while withdrawing support for the Research Works Act, we will continue to join with those many other nonprofit and commercial publishers and scholarly societies that oppose repeated efforts to extend mandates through legislation.

I don’t know what position Springer and Wiley-Blackwell take on this bill: besides Elsevier they’re the biggest science publishers. If they all drop their support, the bill may die. And then we can take the offensive and push for the Federal Research Public Access Act.

This bill would make sure the people who pay for U.S. government-funded research—us, the taxpayers—don’t have to pay again just to see what we bought. It would do this by expanding what’s already standard practice at the National Institute of Health to some other big funding agencies, like the National Science Foundation.

On Google+, open-access hero Peter Suber writes:

This is a victory for what The Economist called the Academic Spring. It shows that academic discontent—expressed in blogs, social media, mainstream news media, and open letters to Congress—can defeat legislation supported by a determined and well-funded lobby. Let’s remember that, and let’s prove that this political force can go beyond defeating bad legislation, like the Research Works Act, to enacting good legislation, like the Federal Research Public Access Act.

Indeed, for companies like Elsevier, the great thing about bills like the Research Work Act is that they make us work hard just to keep the status quo, instead of what we really want: changing the status quo for the better. And they’re perfectly happy to stage a tactical retreat in a little skirmish like this if it distracts us from our real goals.

So, let’s keep at it! For starters, if you teach or study at a university, you can click on the picture below, get a PDF file of a poster that explains the boycott, print it out, and put it on your door. While for some of us the Elsevier boycott is old news, a surprising number of people who should know haven’t heard of it yet!

Luckily, a PR blitz in various math journals will start to change that, at least in the field of mathematics. And soon I’ll talk about some exciting plans being developed on Math 2.0. But if you’re a biologist or chemist, for example, you really need to start the revolution over in your field.

## Elsevier and Springer Sue University Library

20 February, 2012

The battle is heating up! Now Elsevier, Springer and a smaller third publisher are suing a major university in Switzerland, the Eidgenössische Technische Hochschule Zürich, or ETH Zürich for short. Why? Because this university’s library is distributing copies of their journal articles at a lower cost than the publishers themselves.

Aren’t university libraries supposed to make journal articles available? Over on Google+, Willie Wong explains:

My guess is that they are complaining about how the ETH Library (as well as many other libraries in the NEBIS system) offers Electronic Document Delivery.

It is free for staff and researchers, and private individuals who purchase a library membership can ask for articles for a fee. It is a nice service: otherwise most of us would just go to the library, borrow the printed journal, and scan it ourselves (when the electronic copy is not part of the library’s subscription). This way the library does the scanning for us (so we benefit from time better used) and the library benefits from less undesired wear-and-tear and loss from their paper copies.

The publishers probably think the library is illegally reselling their journal articles! But here’s an article by the head of the ETH library, making his side of the case:

• Wolfram Neubauer, A thorn in the side for science publishers, ETH Life, 17 February 2012.

He says the delivery of electronic copies of documents is allowed by the Swiss Copyright Act. He also makes a broader moral case:

• More or less all scientifically relevant journals rely on the results of publicly funded research.

• The brunt of evaluating scientific findings (i.e. peer reviewing) is borne by the scientific community, with the publishers playing only a supporting role.

• By far the most important customers for all major science publishers are academic libraries, the vast majority of which are themselves supported by public funding.

He concludes:

In the legal proceedings, the aim must therefore be to strike a balance between the services provided by the ETH-Bibliothek for the benefit of science and research on the one hand and the commercial interests of the publishers on the other.

It’ll be interesting to see how this goes in court. Either way, a kind of precedent will be set.

## Math 2.0

16 February, 2012

Building on the Elsevier boycott, a lot of people are working on positive steps to make expensive journals obsolete. My email is flooded with discussions, different groups making different plans.

Email is great, but not for everything. So Andrew Stacey (the technical mastermind behind the nLab, Azimuth Wiki and Azimuth Forum):

and Scott Morrison (one of the brains behind MathOverflow, an important math question-and-answer website):

have started a forum to talk about the many issues involved:

Math 2.0.

That’s good, because these guys actually do stuff, not just talk! Andrew describes the idea here:

The purpose of Math 2.0 is to provide a forum for discussion of the future of mathematical publishing. It’s something that I’ve viewed as an important issue for years, and have had many, many interesting conversations about, but somehow nothing much seems to happen. I’m hoping that the momentum from Tim Gowers’ recent blog posts might lead to something and I’d like to capitalise on that.

However, most of the discussion currently is happening in the comments on blog posts. This is hard to follow, and hard to separate out the new suggestions from the discussions on old ones. I think that forums are much better for discussion, hence this one.

The name, Math2.0, is intended to signify two things: that it’s time for an upgrade of the mathematical environment and that I think we can learn a lot from looking at how software—particularly open source software—works. By “mathematical environment”, I don’t mean how we actually do the mathematics but what happens next, particularly communicating the ideas that we create. This is where the internet can really change things for the better (as it has started to do with the arXiv), but where I think that we have yet to figure out how to make best use of it.

This doesn’t just include journals, but I think that that’s an obvious place to start.

So: welcome to Math2.0. Please join in. It’s important.

Andrew Stacey has also emphasized a principle that’s good for reducing chat about starry-eyed visions and focusing on what we can do now:

In all these discussions, there is one point that I would like to make at the start and which I think is relevant to any proposal to set up something new for mathematicians (or more generally, for academics). That is that whatever system is set up it must be:

Useful at the point of use

This is something that I’ve learnt from administering the nLab over the past few years. It keeps going and there is no sign of it slowing down. The secret of its success, I maintain, is that it is useful at the point of use. When I write something on the nLab, I benefit immediately. I can link to previous things I’ve written, to definitions that others have written, and so link my ideas to many others. It means that if I want to talk to someone about something, the thing we are talking about is easily visible and accessible to both (or all) of us. If I want to remember what it was I was thinking about a year ago, I can easily find it. The fact that when I come back the next day, whatever I’ve added has been improved, polished, and added to, is a bonus—but it would still be useful if that didn’t happen.

For other things, then I need more of an incentive to participate. MathOverflow was a lot of fun in the beginning, but now I find that a question needs to be such that it’s fairly clear that I’m one of the few people in the world who can answer. It’s not that my enthusiasm for the site has gone down, just that everything else keeps pushing it out of the way. So a new system has to be useful to those who use it, and ideally the usefulness should be proportional to the amount of effort that one puts in.

A corollary of this is that it should be useful even if only a small number of people use it. The number of core users of the nLab is not large, but nevertheless the nLab is still extremely useful to us. I can imagine that when a proposal for something new is made, there will be a variety of reactions ranging from “That’ll never work” through to “Sounds interesting, but …” with only a few saying “Count me in!”. To have a chance of succeeding, it has to be the case that those few can get it off the ground and demonstrate that it works, without the input of the wider sceptical community.

So: if you’re a mathematician or programmer interested in revolutionizing the future of math publishing, go to Math 2.0, register, and join the conversation! You’ll see there are a number of concrete proposals on the table, including one by Chris Lee, and Marc Harper and myself.

I’ll say more about those later. But I want to add a principle of my own to Andrew’s ‘useful at the point of use’. The goal is not to get a universal consensus on the future of math publishing! Instead, we need a healthy dissensus in which different groups of people develop different systems—so we can see which ones work.

In biology, evolution happens when some change is useful at the point of use—and it doesn’t happen by consensus, either. When some fish gradually became amphibians, they didn’t wait for all fish to agree this was a good move. And indeed it’s good that we still have fish.

Jan Velterop has some interesting thoughts on the evolution of scholarly publishing, which you can read here:

• Richard Poynder, The open access interviews: Jan Velterop, February 2012.

Velterop writes:

As a geologist I go so far as to say that I see analogies with the Permian-Triassic boundary and the Cretaceous-Tertiary boundary, when life on Earth changed dramatically due to fundamental and sudden changes in the environment.

Those boundary events, as they are known, resulted in mass extinctions, and that’s an unavoidable evolutionary consequence of sudden dramatic environmental changes.

But they also open up ecological niches for new, or hitherto less successful, forms of life. In this regard, it is interesting to see the recent announcement of F1000 Research, which intends to address the major issues afflicting scientific publishing.

[…]

The evolution of scientific communication will go on, without any doubt, and although that may not mean the total demise of the traditional models, these models will necessarily change. After all, some dinosaur lineages survived as well. We call them birds. And there are some very attractive ones. They are smaller than the dinosaurs they evolved from, though. Much smaller.

## The Federal Research Public Access Act

10 February, 2012

As of this minute, 5030 scholars have joined the Elsevier boycott. You should too! But now is the time to go further and take positive steps to develop new, better systems for refereeing and distributing scholarly papers.

Everyone I know is talking about this now. Today, quantum physicist Steve Flammia pointed out to me that U.S. Representative Mike Doyle has a good idea:

It’s simple: we should get to see the research we paid for with our tax dollars. We shouldn’t have to pay for it twice: once to have it done, and once more to see the results.

As Doyle puts it:

Americans have the right to see the results of research funded with taxpayer dollars. Yet such research too often gets locked away behind a pay-wall, forcing those who want to learn from it to pay expensive subscription fees for access.

The Federal Research Public Access Act will encourage broader collaboration among scholars in the scientific community by permitting widespread dissemination of research findings. Promoting greater collaboration will inevitably lead to more innovative research outcomes and more effective solutions in the fields of biomedicine, energy, education, and health care.

But what does the bill actually do? It says this: any federal agency that spends more than $100 million per year funding research must make that research freely available in a public repository no later than 6 months after the research has been published in a peer-review journal. This is already done by the National Institute of Health: the bill would expand this practice to the National Science Foundation, the Department of Energy, and other agencies. ### What we should do Someone with technical brains should make it easy for US citizens to contact Congress and support this bill. Google got 4.5 million people to sign their petition against SOPA, the so-called Stop Online Piracy Act. But we’ve been playing defense for too long. Let’s go on the offense and do something like this for a bill that’s good! Emailing your congressperson incredibly easy, but telephone calls are even better, precisely because they’re a bit more work. Here’s a sample of what you could write or say: I am your constituent, and I urge you to support the Federal Research Public Access Act. As a taxpayer, I help support scientific research out of my own pocket. I deserve to see the results! The National Institute of Health already demands this for all the research they support, and the system works well. Broadening this policy will advance science and improve the lives and welfare of all Americans. I believe an emphasis on ‘taxpayers getting their money worth’ and ‘improving the lives of all Americans’ may resonate well with the U.S. Congress: that’s why I’ve worded the message this way. Taxes and patriotism are hot-button issues. But of course you should feel free to modify this text! ### Why it’s important I think this bill is important: even if it doesn’t pass, it changes the debate and puts the publishers on the defensive. Remember: the Association of American Publishers is still supporting the Research Works Act, a bill that would prevent federal agencies from requiring that the research they fund be made freely available online. It seems this bill would even roll back the existing requirement that research funded by the National Institute of Health be made freely available at PubMed Central! There’s a built-in imbalance at work here. Publishers pays lobbyists to work full-time on advancing their agenda. Scientists and other scholars prefer to spend their time thinking about more interesting things. So, we’re usually reactive: we wait until something becomes intolerable before taking action. That’s why we’re fighting against a crisis of journal prices that bankrupt our libraries, and battling bad bills like the Research Works Act, when we should be developing better systems for communicating the results of our research, and supporting good bills… … like the Federal Research Public Access Act! ### For more For more, see: • David Dobbs, Open science revolt occupies Congress, Wired, 9 February 2012. Call to action: Tell Congress you support the Bipartisan Federal Research Public Access Act (FRPAA), Alliance for Taxpayer Access, 9 February 2012. • Scholarly Publishing & Academic Resources Council, SPARC FAQ for university administrators and faculty: Federal Research Public Access Act (FRPAA). The original sponsors of the Federal Research Public Access Act were Reps. Kevin Yoder (R-KS) and Wm. Lacy Clay (D-MO). Identical legislation is also being introduced in the U.S. Senate by Sens. John Cornyn (R-TX), Ron Wyden (D-OR), and Kay Bailey Hutchison (R-TX). ## The Cost of Knowledge 8 February, 2012 As of this moment, 4760 scholars have joined a boycott of the publishing company Elsevier. Of these, only 20% are mathematicians. But since the boycott was started by a mathematician, 34 of us wrote and signed an official statement explaining the boycott: It’s also below. Please check it out and join the boycott! I’m sure more than 34 mathematicians would be happy to sign, but we wanted to get the statement out soon. ### THE COST OF KNOWLEDGE This is an attempt to describe some of the background to the current boycott of Elsevier by many mathematicians (and other academics) at http://thecostofknowledge.com, and to present some of the issues that confront the boycott movement. Although the movement is anything but monolithic, we believe that the points we make here will resonate with many of the signatories to the boycott. #### The role of journals (1): dissemination of research. The role of journals in professional mathematics has been under discussion for some time now. Traditionally, while journals served several purposes, their primary purpose was the dissemination of research papers. The journal publishers were charging for the cost of typesetting (not a trivial matter in general before the advent of electronic typesetting, and particularly non-trivial for mathematics), the cost of physically publishing copies of the journals, and the cost of distributing the journals to subscribers (primarily academic libraries). The editorial board of a journal is a group of professional mathematicians. Their editorial work is undertaken as part of their scholarly duties, and so is paid for by their employer, typically a university. Thus, from the publisher’s viewpoint the editors are volunteers. (The editor in chief of a journal sometimes receives modest compensation from the publisher.) When a paper is submitted to the journal, by an author who is again typically a university-employed mathematician, the editors select the referee or referees for the paper, evaluate the referees’ reports, decide whether or not to accept the submission, and organize the submitted papers into volumes. These are passed on to the publisher, who then undertakes the job of actually publishing them. The publisher supplies some administrative assistance in handling the papers, as well as some copy-editing assistance, which is often quite minor but sometimes more substantial. The referees are again volunteers from the point of view of the publisher: as with editing, refereeing is regarded as part of the service component of a mathematician’s academic work. Authors are not paid by the publishers for their published papers, although they are usually asked to sign over the copyright to the publisher. This system made sense when the publishing and dissemination of papers was a difficult and expensive undertaking. Publishers supplied a valuable service in this regard, for which they were paid by subscribers to the journals, which were mainly academic libraries. The academic institutions whose libraries subscribe to mathematics journals are broadly speaking the same institutions that employ the mathematicians who are writing for, refereeing for, and editing the journals. Therefore, the cost of the whole process of producing research papers is borne by these institutions (and the outside entities that partially fund them, such as the National Science Foundation in the United States): they pay for their academic mathematician employees to do research and to organize the publications of the results of their research in journals; and then (through their libraries) they pay the publishers to disseminate these results among all the world’s mathematicians. Since these institutions employ research faculty in order to foster research, it certainly used to make sense for them to pay for the dissemination of this research as well. After all, the sharing of scientific ideas and research results is unquestionably a key component for making progress in science. Now, however, the world has changed in significant ways. Authors typeset their own papers, using electronic typesetting. Publishing and distribution costs are not as great as they once were. And most importantly, dissemination of scientific ideas no longer takes place via the physical distribution of journal volumes. Rather, it takes place mainly electronically. While this means of dissemination is not free, it is much less expensive, and much of it happens quite independently of mathematical journals. In conclusion, the cost of journal publishing has gone down because the cost of typesetting has been shifted from publishers to authors and the cost of publishing and distribution is significantly lower than it used to be. By contrast, the amount of money being spent by university libraries on journals seems to be growing with no end in sight. Why do mathematicians contribute all this volunteer labor, and their employers pay all this money, for a service whose value no longer justifies its cost? #### The role of journals (2): peer review and professional evaluation There are some important reasons that mathematicians haven’t just abandoned journal publishing. In particular, peer review plays an essential role in ensuring the correctness and readability of mathematical papers, and publishing papers in research journals is the main way of achieving professional recognition. Furthermore, not all journals count equally from this point of view: journals are (loosely) ranked, so that publications in top journals will often count more than publications in lower ranked ones. Professional mathematicians typically have a good sense of the relative prestige of the journals that publish papers in their area, and they will usually submit a paper to the highest ranked journal that they judge is likely to accept and publish it. Because of this evaluative aspect of traditional journal publishing, the problem of switching to a different model is much more difficult than it might appear at first. For example, it is not easy just to begin a new journal (even an electronic one, which avoids the difficulties of printing and distribution), since mathematicians may not want to publish in it, preferring to submit to journals with known reputations. Secondly, although the reputation of various journals has been created through the efforts of the authors, referees, and editors who have worked (at no cost to the publishers) on it over the years, in many cases the name of the journal is owned by the publisher, making it difficult for the mathematical community to separate this valuable object that they have constructed from its present publisher. #### The role of Elsevier Elsevier, Springer, and a number of other commercial publishers (many of them large companies but less significant for their mathematics publishing, e.g., Wiley) all exploit our volunteer labor to extract very large profits from the academic community. They supply some value in the process, but nothing like enough to justify their prices. Among these publishers, Elsevier may not be the most expensive, but in the light of other factors, such as scandals, lawsuits, lobbying, etc. (discussed further below), we consider them a good initial focus for our discontent. A boycott should be substantial enough to be meaningful, but not so broad that the choice of targets becomes controversial or the boycott becomes an unmanageable burden. Refusing to submit papers to all overpriced publishers is a reasonable further step, which some of us have taken, but the focus of this boycott is on Elsevier because of the widespread feeling among mathematicians that they are the worst offender. Let us begin with the issue of journal costs. Unfortunately, it is difficult to make cost comparisons: journals differ greatly in quality, in number of pages per volume, and even in amount of text per page. As measured by list prices, Elsevier mathematics journals are amongst the most expensive. For instance, in the AMS mathematics journal price survey, seven of the ten most expensive journals (by 2007 volume list price) were published by Elsevier. (All prices are as of 2007 because both prices and page counts are easily available online.) However, that is primarily because Elsevier publishes the largest volumes. Price per page is a more meaningful measure that can be easily computed. By this standard, Elsevier is certainly not the worst publisher, but its prices do on the face of it look very high. The Annals of Mathematics, published by Princeton University Press, is one of the absolute top mathematics journals and quite affordably priced:$0.13/page as of 2007. By contrast, ten Elsevier journals (not including one that has since ceased publication) cost $1.30/page or more; they and three others cost more per page than any journal published by a university press or learned society. For comparison, three other top journals competing with the Annals are Acta Mathematica, published by the Institut Mittag Leffler for$0.65/page, Journal of the American Mathematical Society, published by the American Mathematical Society for $0.24/page, and Inventiones Mathematicae, published by Springer for$1.21/page. Note that none of Elsevier’s mathematics journals is generally considered comparable in quality to these journals.

However, there is an additional aspect which makes it hard to compute the true cost of mathematics journals. This is the widespread practice among large commercial publishers of “bundling” journals, which allows libraries to subscribe to large numbers of journals in order to avoid paying the exorbitant list prices for the ones they need. Although this means that the average price libraries pay per journal is less than the list prices might suggest, what really matters is the average price that they pay per journal (or page of journal) that they actually want, which is hard to assess, but clearly higher. We would very much like to be able to offer more concrete data regarding the actual costs to libraries of Elsevier journals compared with those of Springer or other publishers. Unfortunately, this is difficult, because publishers often make it a contractual requirement that their institutional customers should not disclose the financial details of their contracts. For example, Elsevier sued Washington State University to try to prevent release of this information. One common consequence of these arrangements, though, is that in many cases a library cannot actually save any money by cancelling a few Elsevier journals: at best the money can sometimes be diverted to pay for other Elsevier subscriptions.

One reason for focusing on Elsevier rather than, say, Springer is that Springer has had a rich and productive history with the mathematical community. As well as journals, it has published important series of textbooks, monographs, and lecture notes; one could perhaps regard the prices of its journals as a means of subsidizing these other, less profitable, types of publications. Although all these types of publications have become less important with the advent of the internet and the resulting electronic distribution of texts, the long and continuing presence of Springer in the mathematical world has resulted in a store of goodwill being built up in the mathematical community towards them. This store is being rapidly depleted, but has not yet reached zero. See for instance the recent petition to Springer by a number of French mathematicians and departments.

Elsevier does not have a comparable tradition of involvement in mathematics publishing. Many of the mathematics journals that it publishes have been acquired comparatively recently as it has bought up other, smaller publishers. Furthermore, in recent years it has been involved in various scandals regarding the scientific content, or lack thereof, of its journals. One in particular involved the journal Chaos, Solitons & Fractals, which, at the time the scandal broke in 2008–2009, was one of the highest impact factor mathematics journals that Elsevier published. (Elsevier currently reports the five-year impact factor of this journal at 1.729. For sake of comparison, Advances in Mathematics, also published by Elsevier, is reported as having a five-year impact factor of 1.575.) It turned out that the high impact factor was at least partly the result of the journal publishing many papers full of mutual citations. (See Arnold for more information on this and other troubling examples that show the limitations of bibliometric measures of scholarly quality.) Furthermore, Chaos, Solitons & Fractals published many papers that, in our professional judgement, have little or no scientific merit and should not have been published in any reputable journal.

In another notorious episode, this time in medicine, for at least five years Elsevier “published a series of sponsored article compilation publications, on behalf of pharmaceutical clients, that were made to look like journals and lacked the proper disclosures”, as noted by the CEO of Elsevier’s Health Sciences Division.

Recently, Elsevier has lobbied for the Research Works Act, a proposed U.S. law that would undo the National Institutes of Health’s public access policy, which guarantees public access to published research papers based on NIH funding within twelve months of publication (to give publishers time to make a profit). Although most lobbying occurs behind closed doors, Elsevier’s vocal support of this act shows their opposition to a popular and effective open access policy.

These scandals, taken together with the bundling practices, exorbitant prices, and lobbying activities, suggest a publisher motivated purely by profit, with no genuine interest in or commitment to mathematical knowledge and the community of academic mathematicians that generates it. Of course, many Elsevier employees are reasonable people doing their best to contribute to scholarly publishing, and we bear them no ill will. However, the organization as a whole does not seem to have the interests of the mathematical community at heart.

#### The boycott

Not surprisingly, many mathematicians have in recent years lost patience with being involved in a system in which commercial publishers make profits based on the free labor of mathematicians and subscription fees from their institutions’ libraries, for a service that has become largely unnecessary. (See Scott Aaronson’s scathing but all-too-true satirical description of the publishers’ business model.) Among all the commercial publishers, the behavior of Elsevier seemed to many to be the most egregious, and a number of mathematicians had made personal commitments to avoid any involvement with Elsevier journals. (Some journals were also successfully moved from Elsevier to other publishers; e.g., Annales Scientifiques de l’école Normale Supérieure which until recent years was published by Elsevier, is now published by the Société Mathématique de France.)

One of us (Timothy Gowers) decided that it might be useful to
publicize his own personal boycott of Elsevier, thus encouraging others to do the same. This led to the current boycott movement at http://thecostofknowledge.com, the success of which has far exceeded his initial expectations.

Each participant in the boycott can choose which activities they intend to avoid: submitting to Elsevier journals, refereeing for them, and serving on editorial boards. Of course, submitting papers and editing journals are purely voluntary activities, but refereeing is a more subtle issue. The entire peer review system depends on the availability of suitable referees, and its success is one of the great traditions of science: refereeing is felt to be both a burden and an honor, and practically every member of the community willingly takes part in it. However, while we respect and value this tradition, many of us do not wish to see our labor used to support Elsevier’s business model.

#### What next?

As suggested at the very beginning, different participants in the boycott have different goals, both in the short and long term. Some people would like to see the journal system eliminated completely and replaced by something else more adapted to the internet and the possibilities of electronic distribution. Others see journals as continuing to play a role, but with commercial publishing being replaced by open access models. Still others imagine a more modest change, in which commercial publishers are replaced by non-profit entities such as professional societies (e.g., the American Mathematical Society, the London Mathematical Society, and the Société Mathématique de France, all of which already publish a number of journals) or university presses; in this way the value generated by the work of authors, referees, and editors would be returned to the academic and scientific community. These goals need not be mutually exclusive: the world of mathematics journals, like the world of mathematics itself, is large, and open access journals can coexist with traditional journals, as well as with other, more novel means of dissemination and evaluation.

What all the signatories do agree on is that Elsevier is an exemplar of everything that is wrong with the current system of commercial publication of mathematics journals, and we will no longer acquiesce to Elsevier’s harvesting of the value of our and our colleagues’ work.

What future do we envisage for all the papers that would
otherwise be published in Elsevier journals? There are many
other journals being published; perhaps they can pick up at
least some of the slack. Many successful new journals have been founded in recent years, too, including several that are electronic (thus completely eliminating printing and physical distribution costs), and no doubt more will follow. Finally, we hope that the mathematical community will be able to reclaim for itself some of the value that it has given to Elsevier’s journals by moving some of these journals (in name, if possible, and otherwise in spirit) from Elsevier to other publishers. One notable example is the August 10, 2006 resignation of the entire editorial board of the Elsevier journal Topology and their founding of the Journal of Topology, owned by the London Mathematical Society.

None of these changes will be easy; editing a journal is hard work, and founding a new journal, or moving and relaunching an existing journal, is even harder. But the alternative is to continue with the status quo, in which Elsevier harvests ever larger profits from the work of us and our colleagues, and this is both unsustainable and unacceptable.

Signed by:

Scott Aaronson
Massachusetts Institute of Technology

Douglas N. Arnold
University of Minnesota

Artur Avila
IMPA and Institut de Mathématiques de Jussieu

John Baez
University of California, Riverside

Folkmar Bornemann
Technische Universität München

Danny Calegari
Caltech/Cambridge University

Henry Cohn
Microsoft Research New England

Jordan Ellenberg

Matthew Emerton
University of Chicago

Marie Farge
École Normale Supérieure Paris

David Gabai
Princeton University

Timothy Gowers
Cambridge University

Ben Green
Cambridge University

Martin Grötschel
Technische Universität Berlin

Michael Harris
Université Paris-Diderot Paris 7

Frédéric Hélein
Institut de Mathéatiques de Jussieu

Rob Kirby
University of California, Berkeley

Vincent Lafforgue
CNRS and Université d’Orléans

Gregory F. Lawler
University of Chicago

Randall J. LeVeque
University of Washington

László Lovász
Eötvös Lor´nd University

Peter J. Olver
University of Minnesota

Queen Mary, University of London

Terence Tao
University of California, Los Angeles

Richard Taylor

Bernard Teissier
Institut de Mathématiques de Jussieu

Burt Totaro
Cambridge University

Lloyd N. Trefethen
Oxford University

Takashi Tsuboi
University of Tokyo

Marie-France Vigneras
Institut de Mathématiques de Jussieu

Wendelin Werner
Université Paris-Sud

Amie Wilkinson
University of Chicago

Günter M. Ziegler
Freie Universität Berlin

#### Appendix: recommendations for mathematicians.

All mathematicians must decide for themselves whether, or to what extent, they wish to participate in the boycott. Senior
mathematicians who have signed the boycott bear some
responsibility towards junior colleagues who are forgoing the
option of publishing in Elsevier journals, and should do their
best to help minimize any negative career consequences.

Whether or not you decide to join the boycott, there are some
simple actions that everyone can take, which seem to us to be
uncontroversial:

1) Make sure that the final versions of all your papers, particularly new ones, are freely available online— ideally both on the arXiv. (Elsevier’ electronic preprint policy is unacceptable, because it explicitly does not allow authors to update their papers on the arXiv to incorporate changes made during peer review). When signing copyright transfer forms, we recommend amending them (if necessary) to reserve the right to make the author’s final version of the text available free online from servers such as the arXiv, and on your home page.

2) If you are submitting a paper and there is a choice between an expensive journal and a cheap (or free) journal of the same standard, then always submit to the cheap one.

#### Note

The PDF version of this statement has many useful references not included here.

## The Faculty of 1000

31 January, 2012

As of this minute, 1890 scholars have signed a pledge not to cooperate with the publisher Elsevier. People are starting to notice. According to this Wired article, the open-access movement is “catching fire”:

• David Dobbs, Testify: the open-science movement catches fire, Wired, 30 January 2012.

Now is a good time to take more substantial actions. But what?

Many things are being discussed, but it’s good to spend a bit of time thinking about the root problems and the ultimate solutions.

The world-wide web has made journals obsolete: it would be better to put papers on freely available archives and then let boards of top scholars referee them. But how do we get to this system?

In math and physics we have the arXiv, but nobody referees those papers. In biology and medicine, a board called the Faculty of 1000 chooses and evaluates the best papers, but there’s no archive: they get those papers from traditional journals.

Whoops—never mind! That was yesterday. Now the Faculty of 1000 has started an archive!

• Rebecca Lawrence, F1000 Research – join us and shape the future of scholarly communication, F1000, 30 January 2012.

• Ivan Oransky, An arXiv for all of science? F1000 launches new immediate publication journal, Retraction Watch, 30 January 2012.

This blog article says “an arXiv for all science”, but it seems the new F1000 Research archive is just for biology and medicine. So now it’s time for the mathematicians and physicists to start catching up.

## Ban Elsevier

26 January, 2012

Please take the pledge not to do business with Elsevier. 404 scientists have done it so far:

You can separately say you

1) won’t publish with them,
2) won’t referee for them, and/or
3) won’t do editorial work for them.

At least do number 2): how often can you do something good by doing less work? When a huge corporation relies so heavily on nasty monopolistic practices and unpaid volunteer labor, they leave themselves open to this.

This pledge website is the brainchild of Tim Gowers, a Fields medalist and prominent math blogger:

• Tim Gowers, Elsevier: my part in its downfall and http://thecostofknowledge.com.

In case you’re not familiar with the Elsevier problem, here’s something excerpted from my website. This does not yet mention Elsevier’s recent support of the Research Works Act, which would try to roll back the US government’s requirement that taxpayer-funded medical research be made freely available online. Nor does it mention the fake medical journals created by Elsevier, where what looked like peer-reviewed papers were secretly advertisements paid for by drug companies! Nor does it mention the Chaos, Solitons and Fractals fiasco. Indeed, it’s hard keeping up with Elsevier’s dirty deeds!

### The problem and the solutions

The problem of highly priced science journals is well-known. A wave of mergers in the publishing business has created giant firms with the power to extract ever higher journal prices from university libraries. As a result, libraries are continually being forced to cough up more money or cut their journal subscriptions. It’s really become a crisis.

Luckily, there are also two counter-trends at work. In mathematics and physics, more and more papers are available from a free electronic database called the arXiv, and journals are beginning to let papers stay on this database even after they are published. In the life sciences, PubMed Central plays a similar role.

There are also a growing number of free journals. Many of these are peer-reviewed, and most are run by academics instead of large corporations.

The situation is worst in biology and medicine: the extremely profitable spinoffs of research in these subjects has made it easy for journals to charge outrageous prices and limit the free nature of discourse. A non-profit organization called the Public Library of Science was formed to fight this, and circulated an open letter calling on publishers to adopt reasonable policies. 30,000 scientists signed this and pledged to:

publish in, edit or review for, and personally subscribe to only those scholarly and scientific journals that have agreed to grant unrestricted free distribution rights to any and all original research reports that they have published, through PubMed Central and similar online public resources, within 6 months of their initial publication date.

Unsurprisingly, the response from publishers was chilly. As a result, the Public Library of Science started its own free journals in biology and medicine, with the help of a 9 million dollar grant from the Gordon and Betty Moore Foundation.

A number of other organizations are also pushing for free access to scholarly journals, such as Create Change, the Scholarly Publishing and Academic Resources Coalition, and the Budapest Open Access Initiative, funded by George Soros.

Editorial boards are beginning to wise up, too. On August 10, 2006, all the editors of the math journal Topology resigned to protest the outrageous prices of the publisher, Reed Elsevier. In August of this year, the editorial board of the Springer journal K-Theory followed suit. The Ecole Normale Superieure has also stopped having Elsevier publish the journal Annales Scientifiques de l’École Normale Supérieure.

So, we may just win this war! But only if we all do our part.

### What we can do

What can we do to keep academic discourse freely available to all? Here are some things:

1. Don’t publish in overpriced journals.

2. Don’t do free work for overpriced journals (like refereeing and editing).

3. Put your articles on the arXiv or a similar site before publishing them.

4. Only publish in journals that let you keep your articles on the arXiv or a similar site.

5. Support free journals by publishing in them, refereeing for them, editing them… even starting your own!

6. Help make sure free journals and the arXiv stay free.

7. Help start a system of independent ‘referee boards‘ for arXiv papers. These can referee papers and help hiring, tenure and promotion committees to assess the worth of papers, eliminating the last remaining reasons for the existence of traditional for-profit journals.

The nice thing is that most of these are easy to do! Only items 5 through 7 require serious work. As for item 4, a lot of math and physics journals not only let you keep your article on the arXiv, but let you submit it by telling them its arXiv number! In math it’s easy to find these journals, because there’s a public list of them.

Of course, you should read the copyright agreement that you’ll be forced to sign before submitting to a journal or publishing a book. Check to see if you can keep your work on the arXiv, on your own website, etcetera. You can pretty much assume that any rights you don’t explicitly keep, your publisher will get. Eric Weisstein didn’t do this, and look what happened to him: he got sued and spent over a year in legal hell!

Luckily it’s not hard to read these copyright agreements: you can get them off the web. An extensive list is available from Sherpa, an organization devoted to free electronic archives.

If you think maybe you want to start your own journal, or move an existing journal to a cheaper publisher, read Joan Birman’s article about this. Go to the Create Change website and learn what other people are doing. Also check out SPARC—the Scholarly Publishing and Academic Resources Coalition. They can help. And try the Budapest Open Access Initiative—they give out grants.

You can also support the Public Library of Science or join the Open Archives Initiative.

Also: if you like mathematics, tell your librarian about Mathematical Sciences Publishers, a nonprofit organization run by mathematicians for the purpose of publishing low-cost, high-quality math journals.

### Which journals are overpriced?

In 1997 Robion Kirby urged mathematicians not to submit papers to, nor edit for, nor referee for overpriced journals. I think this suggestion is great, and it applies not just to mathematics but all disciplines. There is really no good reason for us to donate our work to profit-making corporations who sell it back to us at exorbitant prices! Indeed in climate science this has a terrible effect: crackpot bloggers distribute their misinformation free of charge, while lots of important serious climate science papers are hidden, available only to people who work at institutions with expensive subscriptions.

But how can you tell if a journal is overpriced? In mathematics, Up-to-date information on the rise of journal prices is available from the American Mathematical Society. They even include an Excel spreadsheet that lets you do your own calculations with this data! Some of this information is nicely summarized on a webpage by Ulf Rehmann. Using these tools you can make up your own mind which journals are too expensive to be worth supporting with your free volunteer labor.

What about other subjects? I don’t know. Maybe you do?

When I first learned how bad the situation was, I started by boycotting all journals published by Reed Elsevier. This juggernaut was formed by merger of Reed Publishing and Elsevier Scientific Press in 1993. In August 2001 it bought Harcourt Press—which in turn owned Academic Press, which ran a journal I helped edit, Advances in Mathematics. I don’t work for that journal anymore! The reason is that Reed Elsevier is a particularly bad culprit when it comes to charging high prices. You can see this from the above lists of journal prices, and you can also see it in the business news. In 2002, Forbes magazine wrote:

If you are not a scientist or a lawyer, you might never guess which company is one of the world’s biggest in online revenue. Ebay will haul in only $1 billion this year. Amazon has$3.5 billion in revenue but is still, famously, losing money. Outperforming them both is Reed Elsevier, the London-based publishing company. Of its $8 billion in likely sales this year,$1.5 billion will come from online delivery of data, and its operating margin on the internet is a fabulous 22%.

Credit this accomplishment to two things. One is that Reed primarily sells not advertising or entertainment but the dry data used by lawyers, doctors, nurses, scientists and teachers. The other is its newfound marketing hustle: Its CEO since 1999 has been Crispin Davis, formerly a soap salesman.

But Davis will have to keep hustling to stay out of trouble. Reed Elsevier has fat margins and high prices in a business based on information—a commodity, and one that is cheaper than ever in the internet era. New technologies and increasingly universal access to free information make it vulnerable to attack from below. Today pirated music downloaded from the web ravages corporate profits in the music industry. Tomorrow could be the publishing industry’s turn.

Some customers accuse Reed Elsevier of price gouging. Daniel DeVito, a patent lawyer with Skadden, Arps, Slate, Meagher & Flom, is a fan of Reed’s legal-search service, but he himself does free science searches on the Google site before paying for something like Reed’s ScienceDirect—and often finds what he’s looking for at no cost. Reed can ill afford to rest.

Why should we slave away unpaid to keep Crispin Davis and his ilk rolling in dough? There’s really no good reason.

### Sneaky tricks

To fight against the free journals and the arXiv, publishing companies are playing sneaky tricks like these:

Proprietary Preprint Archives. Examples included ChemWeb and something they called "The Mathematics Preprint Server". The latter was especially devious, because mathematicians used to call the arXiv "the mathematics preprint server".

However, the Mathematics Preprint Server didn’t fool many smart people, so lots of the papers they got were crap, like a supposed proof of Goldbach’s conjecture, and a claim that the rotation of a galactic supercluster is due to a "topological defect" in spacetime. Eventually Elsevier gave up and stopped accepting new papers on their preprint server. Now it’s a laughable shadow of its former self. Similarly, ChemWeb was sold off.

Web Spamming. More recently, publishers have tried a new trick: “web spamming”, also known as “search engine spamming” or “cloaking”. The company gives search engine crawlers access to full-text articles — but when you try to read these articles, you get a "doorway page" demanding a subscription or payment. Sometimes you’ll even be taken to a page that has nothing to do with the paper you thought you were about to see!

Culprits include Springer, Reed Elsevier, and the Institute of Electrical and Electronic Engineers. The last one seems to have quit — but check out their powerpoint presentation on this subject, courtesy of Carl Willis.

If you see pages like this, report them to Google or your favorite search engine.

Journal Bundling. Worse still is the strategy of "bundling" subscriptions into huge all-or-nothing packages, so libraries can’t save money by ceasing to subscribe to a single journal. It’s a clever trap, especially because these bundled subscriptions look like a good deal at first. The cost becomes apparent only later. Now universities libraries are being bankrupted as the prices of these bundles keep soaring. The library of my own university, U.C. Riverside, barely has money for any books anymore!

Luckily, people are catching on. In 2003, Cornell University bravely dropped their subscription to 930 Elsevier journals. Four North Carolina universities have joined the revolt, and the University of California has also been battling Elsevier. For other actions universities have taken, read Peter Suber’s list.

Legal bullying. Large corporations like to scare people by means of threats of legal action backed up by deep pockets. A classic example is the lawsuit launched by Gordon and Breach against the American Physical Society for publishing lists of journal prices. Luckily they lost this suit.

Hiring a Dr. Evil lookalike as their PR consultant.

Click either of the pictures for an explanation.

## Classical Mechanics versus Thermodynamics (Part 2)

23 January, 2012

I showed you last time that in many branches of physics—including classical mechanics and thermodynamics—we can see our task as minimizing or maximizing some function. Today I want to show how we get from that task to symplectic geometry.

So, suppose we have a smooth function

$S: Q \to \mathbb{R}$

where $Q$ is some manifold. A minimum or maximum of $S$ can only occur at a point where

$d S = 0$

Here the differential $d S$ which is a 1-form on $Q.$ If we pick local coordinates $q^i$ in some open set of $Q,$ then we have

$\displaystyle {d S = \frac{\partial S}{\partial q^i} dq^i }$

and these derivatives $\displaystyle{ \frac{\partial S}{\partial q^i} }$ are very interesting. Let’s see why:

Example 1. In classical mechanics, consider a particle on a manifold $X.$ Suppose the particle starts at some fixed position at some fixed time. Suppose that it ends up at the position $x$ at time $t.$ Then the particle will seek to follow a path that minimizes the action given these conditions. Assume this path exists and is unique. The action of this path is then called Hamilton’s principal function, $S(x,t).$ Let

$Q = X \times \mathbb{R}$

and assume Hamilton’s principal function is a smooth function

$S : Q \to \mathbb{R}$

We then have

$d S = p_i dq^i - H d t$

where $q^i$ are local coordinates on $X,$

$\displaystyle{ p_i = \frac{\partial S}{\partial q^i} }$

is called the momentum in the ith direction, and

$\displaystyle{ H = - \frac{\partial S}{\partial t} }$

is called the energy. The minus signs here are basically just a mild nuisance. Time is different from space, and in special relativity the difference comes from a minus sign, but I don’t think that’s the explanation here. We could get rid of the minus signs by working with negative energy, but it’s not such a big deal.

Example 2. In thermodynamics, consider a system with the internal energy $U$ and volume $V.$ Then the system will choose a state that maximizes the entropy given these constraints. Assume this state exists and is unique. Call the entropy of this state $S(U,V).$ Let

$Q = \mathbb{R}^2$

and assume the entropy is a smooth function

$S : Q \to \mathbb{R}$

We then have

$d S = \displaystyle{\frac{1}{T} d U - \frac{P}{T} d V }$

where $T$ is the temperature of the system, and $P$ is the pressure. The slight awkwardness of this formula makes people favor other setups.

Example 3. In thermodynamics there are many setups for studying the same system using different minimum or maximum principles. One of the most popular is called the energy scheme. If internal energy increases with increasing entropy, as usually the case, this scheme is equivalent to the one we just saw.

In the energy scheme we fix the entropy $S$ and volume $V.$ Then the system will choose a state that minimizes the internal energy given these constraints. Assume this state exists and is unique. Call the internal energy of this state $U(S,V).$ Let

$Q = \mathbb{R}^2$

and assume the entropy is a smooth function

$S : Q \to \mathbb{R}$

We then have

$d U = T d S - P d V$

where

$\displaystyle{ T = \frac{\partial U}{\partial S} }$

is the temperature, and

$\displaystyle{ P = - \frac{\partial U}{\partial V} }$

is the pressure. You’ll note the formulas here closely resemble those in Example 1!

Example 4. Here are the four most popular schemes for thermodynamics:

• If we fix the entropy $S$ and volume $V,$ the system will choose a state that minimizes the internal energy $U(S,V).$

• If we fix the entropy $S$ and pressure $P,$ the system will choose a state that minimizes the enthalpy $H(S,P).$

• If we fix the temperature $T$ and volume $V,$ the system will choose a state that minimizes the Helmholtz free energy $A(T,V).$

• If we fix the temperature $T$ and pressure $P,$ the system will choose a state that minimizes the Gibbs free energy $G(T,P).$

These quantities are related by a pack of similar-looking formulas, from which we may derive a mind-numbing little labyrinth of Maxwell relations. But for now, all we need to know is that all these approaches to thermodynamics are equivalent given some reasonable assumptions, and all the formulas and relations can be derived using the Legendre transformation trick I explained last time. So, I won’t repeat what we did in Example 3 for all these other cases!

Example 5. In classical statics, consider a particle on a manifold $Q.$ This particle will seek to minimize its potential energy $V(q),$ which we’ll assume is some smooth function of its position $q \in Q.$ We then have

$d V = -F_i dq^i$

where $q^i$ are local coordinates on $Q$ and

$\displaystyle{ F_i = -\frac{\partial V}{\partial q^i} }$

is called the force in the ith direction.

#### Conjugate variables

So, the partial derivatives of the quantity we’re trying to minimize or maximize are very important! As a result, we often want to give them more of an equal status as independent quantities in their own right. Then we call them ‘conjugate variables’.

To make this precise, consider the cotangent bundle $T^* Q,$ which has local coordinates $q^i$ (coming from the coordinates on $Q$) and $p_i$ (the corresponding coordinates on each cotangent space). We then call $p_i$ the conjugate variable of the coordinate $q^i.$

Given a smooth function

$S : Q \to \mathbb{R}$

the 1-form $d S$ can be seen as a section of the cotangent bundle. The graph of this section is defined by the equation

$\displaystyle{ p_i = \frac{\partial S}{\partial q^i} }$

and this equation ties together two intuitions about ‘conjugate variables’: as coordinates on the cotangent bundle, and as partial derivatives of the quantity we’re trying to minimize or maximize.

#### The tautological 1-form

There is a lot to say here, especially about Legendre transformations, but I want to hasten on to a bit of symplectic geometry. And for this we need the ‘tautological 1-form’ on $T^* Q.$

We can think of $d S$ as a map

$d S : Q \to T^* Q$

sending each point $q \in Q$ to the point $(q,p) \in T^* Q$ where $p$ is defined by the equation we just saw:

$\displaystyle{ p_i = \frac{\partial S}{\partial q^i} }$

Using this map, we can pull back any 1-form on $T^* Q$ to get a 1-form on $Q.$

What 1-form on $Q$ might we like to get? Why, $d S$ of course!

Amazingly, there’s a 1-form $\alpha$ on $T^* Q$ such that when we pull it back using the map $d S,$ we get the 1-form $d S$—no matter what smooth function $d S$ we started with!

Thanks to this wonderfully tautological property, $\alpha$ is called the tautological 1-form on $T^* Q.$ You should check that it’s given by the formula

$\alpha = p_i dq^i$

If you get stuck, try this.

So, if we want to see how much $S$ changes as we move along a path in $Q,$ we can do this in three equivalent ways:

• Evaluate $S$ at the endpoint of the path and subtract off $S$ at the starting-point.

• Integrate the 1-form $d S$ along the path.

• Use $d S : Q \to T^* Q$ to map the path over to $T^* Q,$ and then integrate $\alpha$ over this path in $T^* Q.$

The last method is equivalent thanks to the ‘tautological’ property of $\alpha.$ It may seem overly convoluted, but it shows that if we work in $T^* Q,$ where the conjugate variables are accorded equal status, everything we want to know about the change in $S$ is contained in the 1-form $\alpha,$ no matter which function $S$ we decide to use!

So, in this sense, $\alpha$ knows everything there is to know about the change in Hamilton’s principal function in classical mechanics, or the change in entropy in thermodynamics… and so on!

But this means it must know about things like Hamilton’s equations, and the Maxwell relations.

#### The symplectic structure

We saw last time that the fundamental equations of classical mechanics and thermodynamics—Hamilton’s equations and the Maxwell relations—are mathematically just the same. They both say simply that partial derivatives commute:

$\displaystyle { \frac{\partial^2 S}{\partial q^i \partial q^j} = \frac{\partial^2 S}{\partial q^j \partial q^i} }$

where $S: Q \to \mathbb{R}$ is the function we’re trying to minimize or maximize.

I also mentioned that this fact—the commuting of partial derivatives—can be stated in an elegant coordinate-free way:

$d^2 S = 0$

Perhaps I should remind you of the proof:

$d^2 S = d \left( \displaystyle{ \frac{\partial S}{\partial q^i} dq^i } \right) = \displaystyle{ \frac{\partial^2 S}{\partial q^j \partial q^i} dq^j \wedge dq^i }$

but

$dq^j \wedge dq^i$

changes sign when we switch $i$ and $j,$ while

$\displaystyle{ \frac{\partial^2 S}{\partial q^j \partial q^i}}$

does not, so $d^2 S = 0.$ It’s just a wee bit more work to show that conversely, starting from $d^2 S = 0,$ it follows that the mixed partials must commute.

How can we state this fact using the tautological 1-form $\alpha$? I said that using the map

$d S : Q \to T^* Q$

we can pull back $\alpha$ to $Q$ and get $d S.$ But pulling back commutes with the $d$ operator! So, if we pull back $d \alpha,$ we get $d^2 S.$ But $d^2 S = 0.$ So, $d \alpha$ has the magical property that when we pull it back to $Q,$ we always get zero, no matter what $S$ we choose!

This magical property captures Hamilton’s equations, the Maxwell relations and so on—for all choices of $S$ at once. So it shouldn’t be surprising that the 2-form

$\theta = d \alpha$

is colossally important: it’s the famous symplectic structure on the so-called phase space $T^* Q.$

Well, actually, most people prefer to work with

$\omega = - d \alpha$

It seems this whole subject is a monument of austere beauty… covered with minus signs, like bird droppings.

Example 6. In classical mechanics, let

$Q = X \times \mathbb{R}$

as in Example 1. If $Q$ has local coordinates $q^i, t,$ then $T^* Q$ has these along with the conjugate variables as coordinates. As we explained, it causes little trouble to call these conjugate variables by the same names we used for the partial derivatives of $S:$ namely, $p_i$ and $-H.$ So, we have

$\alpha = p_i dq^i - H d t$

and thus

$\omega = dq^i \wedge dp_i - dt \wedge dH$

Example 7. In thermodynamics, let

$Q = \mathbb{R}^2$

as in Example 3. If $Q$ has coordinates $S, V$ then the conjugate variables deserve to be called $T, -P.$ So, we have

$\alpha = T d S - P d V$

and

$\omega = d S \wedge d T - d V \wedge d P$

You’ll see that in these formulas for $\omega,$ variables get paired with their conjugate variables. That’s nice.

But let me expand on what we just saw, since it’s important. And let me talk about $\theta = d\alpha,$ without tossing in that extra sign.

What we saw is that the 2-form $\theta$ is a ‘measure of noncommutativity’. When we pull $\theta$ back to $Q$ we get zero. This says that partial derivatives commute—and this gives Hamilton’s equations, the Maxwell relations, and all that. But up in $T^* Q,$ $\theta$ is not zero. And this suggests that there’s some built-in noncommutativity hiding in phase space!

Indeed, we can make this very precise. Consider a little parallelogram up in $T^* Q$:

Suppose we integrate the 1-form $\alpha$ up the left edge and across the top. Do we get the same answer if integrate it across the bottom edge and then up the right?

No, not necessarily! The difference is the same as the integral of $\alpha$ all the way around the parallelogram. By Stokes’ theorem, this is the same as integrating $\theta$ over the parallelogram. And there’s no reason that should give zero.

However, suppose we got our parallelogram in $T^* Q$ by taking a parallelogram in $Q$ and applying the map

$d S : Q \to T^* Q$

Then the integral of $\alpha$ around our parallelogram would be zero, since it would equal the integral of $d S$ around a parallelogram in $Q$… and that’s the change in $S$ as we go around a loop from some point to… itself!

And indeed, the fact that a function $S$ doesn’t change when we go around a parallelogram is precisely what makes

$\displaystyle { \frac{\partial^2 S}{\partial q^i \partial q^j} = \frac{\partial^2 S}{\partial q^j \partial q^i} }$

So the story all fits together quite nicely.

#### The big picture

I’ve tried to show you that the symplectic structure on the phase spaces of classical mechanics, and the lesser-known but utterly analogous one on the phase spaces of thermodynamics, is a natural outgrowth of utterly trivial reflections on the process of minimizing or maximizing a function $S$ on a manifold $Q.$

The first derivative test tells us to look for points with

$d S = 0$

while the commutativity of partial derivatives says that

$d^2 S = 0$

everywhere—and this gives Hamilton’s equations and the Maxwell relations. The 1-form $d S$ is the pullback of the tautologous 1-form $\alpha$ on $T^* Q,$ and similarly $d^2 S$ is the pullback of the symplectic structure $d\alpha.$ The fact that

$d \alpha \ne 0$

says that $T^* Q$ holds noncommutative delights, almost like a world where partial derivatives no longer commute! But of course we still have

$d^2 \alpha = 0$

everywhere, and this becomes part of the official definition of a symplectic structure.

All very simple. I hope, however, the experts note that to see this unified picture, we had to avoid the most common approaches to classical mechanics, which start with either a ‘Hamiltonian’

$H : T^* Q \to \mathbb{R}$

or a ‘Lagrangian’

$L : T Q \to \mathbb{R}$

Instead, we started with Hamilton’s principal function

$S : Q \to \mathbb{R}$

where $Q$ is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the Maxwell relations, become a trivial consequence of the fact that partial derivatives commute.

But what about those ‘noncommutative delights’? First, there’s a noncommutative Poisson bracket operation on functions on $T^* Q.$ This makes the functions into a so-called Poisson algebra. In classical mechanics of a point particle on the line, for example, it’s well-known that we have

$\begin{array}{ccr} \{ p, q \} &=& 1 \\ \{ H, t \} &=& -1 \end{array}$

In thermodynamics, the analogous relations

$\begin{array}{ccr} \{ T, S \} &=& 1 \\ \{ P, V \} &=& -1 \end{array}$

seem sadly little-known. But you can see them here, for example:

• M. J. Peterson, Analogy between thermodynamics and mechanics, American Journal of Physics 47 (1979), 488–490.

at least up to one of those pesky minus signs! We can use these Poisson brackets to study how one thermodynamic variable changes as we slowly change another, staying close to equilibrium all along.

Second, we can go further and ‘quantize’ the functions on $T^* Q.$ This means coming up with an associative but noncommutative product of these function that mimics the Poisson bracket to some extent. In the case of a particle on a line, we’d get commutation relations like

$\begin{array}{lcr} p q - q p &=& - i \hbar \\ H t - t H &=& i \hbar \end{array}$

where $\hbar$ is Planck’s constant. Now we can represent these quantities as operators on a Hilbert space, the uncertainty principle kicks in, and life gets really interesting.

In thermodynamics, the analogous relations would be

$\begin{array}{ccr} T S - S T &=& - i \hbar \\ P V - V P &=& i \hbar \end{array}$

The math works just the same, but what does it mean physically? Are we now thinking of temperature, entropy and the like as ‘quantum observables’—for example, operators on a Hilbert space? Are we just quantizing thermodynamics?

That’s one possible interpretation, but I’ve never heard anyone discuss it. Here’s one good reason: as Blake Stacey pointed out below, these equations don’t pass the test of dimensional analysis! The quantities at left have units of energy, while Plank’s constant has units of action. So maybe we need to introduce a quantity with units of time at right, or maybe there’s some other interpretation, where we don’t interpret the parameter $\hbar$ as the good old-fashioned Planck’s constant, but something else instead.

And if you’ve really been paying attention, you may wonder how quantropy fits into this game! I showed that at least in a toy model, the path integral formulation of quantum mechanics arises, not exactly from maximizing or minimizing something, but from finding its critical points: that is, points where its first derivative vanishes. This something is a complex-valued quantity analogous to entropy, which I called ‘quantropy’.

Now, while I keep throwing around words like ‘minimize’ and ‘maximize’, most everything I’m doing works just fine for critical points. So, it seems that the apparatus of symplectic geometry may apply to the path-integral formulation of quantum mechanics.

But that would be weirdly interesting! In particular, what would happen when we go ahead and quantize the path-integral formulation of quantum mechanics?

If you’re a physicist, there’s a guess that will come tripping off your tongue at this point, without you even needing to think. Me too. But I don’t know if that guess is right.

Less mind-blowingly, there is also the question of how symplectic geometry enters into classical statics via the idea of Example 4.

But there’s a lot of fun to be had in this game already with thermodynamics.

#### Appendix

I should admit, just so you don’t think I failed to notice, that only rather esoteric physicists study the approach to quantum mechanics where time is an operator that doesn’t commute with the Hamiltonian $H.$ In this approach $H$ commutes with the momentum and position operators. I didn’t write down those commutation equations, for fear you’d think I was a crackpot and stop reading! It is however a perfectly respectable approach, which can be reconciled with the usual one. And this issue is not only quantum-mechanical: it’s also important in classical mechanics.

Namely, there’s a way to start with the so-called extended phase space for a point particle on a manifold $X$:

$T^* (X \times \mathbb{R})$

with coordinates $q^i, t, p_i$ and $H,$ and get back to the usual phase space:

$T^* X$

with just $q^i$ and $p_i$ as coordinates. The idea is to impose a constraint of the form

$H = f(q,p)$

to knock off one degree of freedom, and use a standard trick called ‘symplectic reduction’ to knock off another.

Similarly, in quantum mechanics we can start with a big Hilbert space

$L^2(X \times \mathbb{R})$

on which $q^i, t, p_i,$ and $H$ are all operators, then impose a constraint expressing $H$ in terms of $p$ and $q,$ and then use that constraint to pick out states lying in a smaller Hilbert space. This smaller Hilbert space is naturally identified with the usual Hilbert space for a point particle:

$L^2(X)$

Here $X$ is called the configuration space for our particle; its cotangent bundle is the usual phase space. We call $X \times \mathbb{R}$ the extended configuration space for a particle on the line; its cotangent bundle is the extended phase space.

I’m having some trouble remembering where I first learned about these ideas, but here are some good places to start:

• Toby Bartels, Abstract Hamiltonian mechanics.

• Nikola Buric and Slobodan Prvanovic, Space of events and the time observable.

• Piret Kuusk and Madis Koiv, Measurement of time in nonrelativistic quantum and classical mechanics, Proceedings of the Estonian Academy of Sciences, Physics and Mathematics 50 (2001), 195–213.

## Curriki

8 August, 2010

Textbooks are expensive. They could be almost free, especially in subjects like trigonometry or calculus, which don’t change very fast.

I’m a radical when it comes to the dissemination of knowledge: I want to give as much away for free as I can! So if I weren’t doing Azimuth, I’d probably be working to push for open-source textbooks.

Luckily, someone much better at this sort of thing is already doing that. David Roberts — a mathematician you may have seen at the n-Category Café — recently pointed out this good news:

• Ashley Vance, $200 Textbook vs. Free — You Do the Math, New York Times, July 31, 2010. Scott McNealy, cofounder of Sun Microsystems, recently said goodbye to that company and started spearheading a push towards open-source textbooks: Early this year, Oracle, the database software maker, acquired Sun for$7.4 billion, leaving Mr. McNealy without a job. He has since decided to aim his energy and some money at Curriki, an online hub for free textbooks and other course material that he spearheaded six years ago.

“We are spending $8 billion to$15 billion per year on textbooks” in the United States, Mr. McNealy says. “It seems to me we could put that all online for free.”

The nonprofit Curriki fits into an ever-expanding list of organizations that seek to bring the blunt force of Internet economics to bear on the education market. Even the traditional textbook publishers agree that the days of tweaking a few pages in a book just to sell a new edition are coming to an end.

Whenever it happens, it’ll be none too soon for me!

Let us hope that someday the Azimuth Project becomes part of this trend…