Math websites falling into disuse?

Since I recently blogged about Math Overflow website, I’ve been wondering what happened to various other math websites that once looked promising, and how they’re faring. Some of them seem to be going strong, but none of them seem to have been exploding in popularity.

Tricki

I blogged twice about Tricki, the Tricks Wiki, which went live in April 2009 (see the annoucement by Tim Gowers). Tricki held a lot of promise. Of late, the enthusiasm seems to have slowed down, though this might be a temporary phenomenon. The most recently created article and the most recent comments appear to be two weeks old as of today (October 27, 2009). According to Alexa data, the site has a rank of 1,200,000+ worldwide and about 550,000-600,000 in the United States. For comparison, subwiki.org, which I run, has Alexa data showing a site rank of 500,000-550,000 in the world and 150,000-200,000 in the United States, while Math Overflow has Alexa data showing a rank of 350,000 worldwide and about 60,000 in the United States (the numbers you see clicking on the links may be different if you don’t view this post within a few hours of my writing it).

Tricki also hasn’t been mentioned on Gowers’ blog since June 25, 2009 and on Terence Tao’s blog since August 2009.

Is the Tricki falling into disuse? Clearly, the initial spate of interst seems to have subsided, but it might well regain a slower and steadier momentum in some time.

Planetmath

I remember a time when Wikipedia had much less mathematical content than planetmath, which was one of the first places to check mathematics on the Internet. Planetmath appears to be going strong, though not as strong as before. While their message forum seems reasonably active, their latest addition was about a week ago, and they seem to be getting somewhere between 0 and 2 new articles in a day, and around the same number of revisions a day. Not exactly dead, but not bubbling with life. Their Alexa data indicates fairly steady performance with a traffic rank of around 130,000 over the last six months, but a decline over a longer timeframe — setting the drop-down parameter to “max” below the chart shows that their traffic rank and daily pageviews have been following over the longer run. Why? Decline in quality? Probably not — it’s more likely that people are increasingly using Wikipedia.

Are textbooks getting too expensive?

I recently came across a post by John Baez on the n-category cafe titled Cheaper Online Textbooks?. Baez’s post has a number of interesting links: a piece on “Affordable Higher Education” by CALPIRG, a piece on the legislation based on this report by Capital Campus News, an article in the Christan Science Monitor on the rising cost of textbooks, and a blog post in the Chronicle of Higher Education on an e-textbook program. So, reading about all these posts, I began to wonder: are textbooks getting too expensive? And should anything be “done” about it?

Are textbook prices soaring?

So I decided to look at the range of calculus books. The general impression from the things I read seemed to be that it would be hard to get a decent textbook for under $100. So, I went and typed calculus on Amazon, and looked at the first page of search results. Among these search results was Calculus for Dummies ($12.99), Forgotten Calculus ($11.53), Calculus Made Easy ($26.95), Schaum’s Outline of Calculus ($12.89), and The Complete Idiot’s Guide to Calculus ($12.89). Most of these books would seem reasonable for a low-level introductory semester or two quarters in calculus — admittedly, they may not be suitable for all calculus courses, but if price really is a primary consideration, it isn’t as if there are no options. There is also a wikibook on Calculus and an old public domain book on calculus. If you want somewhat more advanced stuff for free, you can try MIT’s OpenCourseWare course on single variable course, which includes video lectures, their course on calculus with applications, and their course on multivariable calculus.

Okay, so perhaps calculus is a bad example? Well, I decided to pick point set topology. The standard book for this is the second edition of Munkres’ book, which I think is one of the best, and it costs $107.73 on Amazon. But searching for topology on Amazon gives a number of other considerably cheaper books, such as Mendelson ($7.88), Gamelin and Greene ($10.17), Springer Undergraduate Math Series book by Crossley ($23.40), Schaum’s Outline of General Topology ($12.89), among many others. None of them seem as good as Munkres, but they all cover the basic material — and reasonably well, it seems.

Of course, I have picked on calculus and topology, both topics that are more than fifty years old, and where most of the material that should be included in an elementary textbook is widely known. In other words, the field for writing books is wide open. No publisher or author has significant scarcity power. When we are looking at exotic topics such as the theory of locally finite groups, then yes, you probably wouldn’t find cheap textbooks. But most undergraduate-level textbooks would likely be of the level of calculus or topology texts, and not exotic texts on locally finite groups.

Why do instructors choose expensive textbooks when cheaper alternatives exist?

Why do instructors choose $100+ calculus textbooks or Munkres’ topology textbook when there are so many cheaper books available on the market? One explanation, pointed out in a comment to Baez’s post, is the “moral hazard” explanation. This states that instructors do not need to bear the costs of buying the textbooks, so they just prescribe the “best” textbook based on their personal criteria rather than taking the price into consideration.

Polymath again

Timothy Gowers and Michael Nielsen have written an article for Nature magazine about the polymath project (I blogged about this here and here).

In the meantime, Terence Tao started a polymath blog here, where he initiated four discussion threads (1, 2, 3 and 4) on deterministic ways to find primes (I’m not quite sure how that’s proceeding — the last post was on August 28, 2009). (UPDATE: A new post (thread 5) was put up shortly after I published my blog post).

Around the same time, Gil Kalai started a polymath on the polynomial Hirsch conjecture (1, 2, 3, 4 and 5).

Also, some general discussion posts on polymath projects: by Tim Gowers and by Terence Tao.

It remains to be seen whether any of these projects are able to reach successful conclusions or make substantial inroads into the problem. If there is another success for a polymath project, then that would be a major booster to the idea of polymath projects. Otherwise, it might raise the question of whether the unexpected degree of success of the first polymath project led by Gowers (which aimed for, and got, an elementary proof of the density Hales-Jewett theorem) was an anomaly.

Math overflow

In recent times, the Math Overflow website has been getting a lot of “press”, which is to say, it has been mentioned in some highly prominent math blogs. It was reviewed in Secret Blogging Seminar by Scott Morrison, who is also involved with Math Overflow, and it was mentioned by quomodocumque, Timothy Gowers, Terence Tao, the n-category cafe and others.

Math Overflow is a website where people can ask math-related questions (the questions should be of interest to people at the level of Ph.D. student or higher), answer the questions, and rate the answers. It uses the Stack Exchange software, which is used for many other websites, such as Stack Overflow. Funding for the website is being provided by Ravi Vakil of Stanford University, and it has a bunch of moderators — but anybody who earns enough points through participation can rise to the status of moderator. For more information, see the Math Overflow FAQ.

Participation on the website has been increasing rapidly since the first post (September 28). Here’s the Alexa data, which seem to indicate that usage has been growing (Alexa is not very reliable for low-volume sites, since it uses a small sample of users and most Math Overflow users may not be using Alexa’s toolbar).

The software and site layout seem well-designed to encourage participation. The long-term performance seems unclear, since a lot depends on how effectively the site is able to allow users to fruitfully explore past questions and answers and discover things similar to what interests them. But, as of now, it has a bunch of interesting questions, and seems to have reached the ears of a lot of people who’re interested in asking good questions and giving good answers.

Collaborative mathematics, etc.

UPDATE: See the polymath project backgrounder for the latest information.

It’s been some time since I last wrote about the polymath project (see this, this for past coverage), and an even longer time since I wrote an extremely lengthy blog post about Michael Nielsen’s ideas about collaborative science.

The first polymath project, polymath1, was about the density Hales-Jewett theorem. This was declared a success, since the original problem was solved within about a month, though the writing up of the paper is still proceeding. The problem for the project was proposed by Timothy Gowers.

Terry Tao (WordPress blog) has now started a polymath blog discussing possible open problems for the polymath project, strategies for how to organize the problem-selection and problem-solving process, and other issues related to writing up final solutions and sharing of credit.

In this blog post, Jon Udell reflects on how the introduction of LaTeX typesetting into wordpress was a positive factor in getting talented mathematicians like Terence Tao and Timothy Gowers into the blogosphere, and leading to innovative user projects such as the polymath project. Udell notes that introducing existing typesetting solutions into new contexts such as Internet blogging software can have profound positive effects.

Information costs and open access

In recent years, there has been a growing trend towards “open access” among librarians and academics. For instance, the University of Michigan recently held an Open Access Week, where they describe open access as:

free, permanent, full-text, online access to peer-reviewed scientific and scholarly material.

In an earlier blog post, I discussed some issues related to open access. Here, I attempt to look at the matter more comprehensively.

Rationales for open access

There are many rationales for open access. The simplest rationale is that open access means reduced cost of access to information, which allows more people to use the same research. Since the marginal cost of making research available via the Internet to more people is near zero, it makes sense from the point of view of efficiency to price access by yet another person to the research at zero.

Another rationale is a more romantic one: making scientific and scholarly publishing available openly allowsfor a free flow of ideas and a grander “conversation”. Support for this rationale also indicates that open access should be more than just free (in the sense of zero cost) access to materials, but also a license that permits liberal reuse of research materials in new contexts. Academia already has strong traditions of quoting from, linking to, and building upon, past work, but this form of open access seeks to provide a legal framework that explicitly specifies reuse rights that go beyond the traditional copyright framework of countries such as the United States. An example of such permissive licensing is the Creative Commons licenses.

As shorthand for these two rationales, I shall use cost rationale and conversation rationale.

Open access policies/mandates

One of the major problems the open access movement has faced so far is getting people to publish papers in open access journals. As long as the best papers continue to be published in closed-access journals, academics who want to read these journals will pressure their university libraries to subscribe to these journals, even when the journals overcharge. Thus, librarians are unable to push open-access terms on publishers.

About “topic pages”: newspapers and others

In a previous post mainly about the Tricki, I conjectured that in order to make information easiest to access, the Tricki should seriously consider the use of topic pages, that serve as central hubs from which users of the Tricki can get an idea of the Tricki’s coverage on a particular topic. I argued that, even in the presence of a website such as Wikipedia that has a high-quality article on the topic (though this is a moot point according to me) such a page serves a useful local function. I also pointed to the example of New York Times Topic pages. See, for instance, the Times Topic pages on Wikipedia, Twitter, and The Amazon Kindle.

Apparently, as I learned from this Register article, Google, in a testimony by its employee Marissa Mayer (PDF), said that:

Consider instead how the authoritativeness of news articles might grow if an evolving story were published under a permanent, single URL as a living, changing, updating entity. We see this practice today in Wikipedia’s entries and in the topic pages at NYTimes.com. The result is a single authoritative page with a consistent reference point that gains clout and a following of users over time.

To paraphrase, I think Mayer’s point is that individual news stories come and go, they are hard to search, find, and catalogue, but a single evolving story can gain clout over time as people refer to it and know it’ll be there. Apparently, another blogger saw a connection between the issue of topic pages for journalists and my suggestion to have topic pages on the Tricki.

Two kinds of users

I have no knowledge about building newspaper websites; in fact, I have extremely limited knowledge about building websites. So what I’m saying here should be treated largely as amateur speculation.

I broadly see two reasons why people visit sites on the Internet. One is to get the new, the latest, information, wherein they don’t know exactly what they’re looking for but want new information. For instance, when I type in a newspaper or periodical website into my browser URL, I’m not looking for specific information, but rather, looking for news. At the other extreme are situations where I am looking for specific information. For instance, I want to know whether there exists an infinite group where every element has finite order.

Of course, much web use lies somewhere in between. For instance, when I follow a subject-specific blog, such as this or this or this, then what I’m looking for is “new” information but restricted to something specific rather than the general wide swath of things that could happen to the world. Something similar may be said for social networking, in so far as the information-gathering there is related to updates and new information but limited to friends and others in one’s network.

On the Internet, these two forms of exploration serve each other. As I read a news article in the New York Times or an editorial in Foreign Policy, there are terms, concepts, ideas, historical facts, etc. that I encounter about which I don’t know enough. I am curious about other coverage of these. In this case, an initial desire for news in general gives rise to specific questions. Conversely, sometimes I get stuck at specific questions, so I do a more general exploration, looking for news that might shed light on one of the many specific questions I have unresolved.

An astute reader might point out that if the main value of “new” information is to provide stimulus for curiosity, the “new” feature isn’t crucial at all. Reading the New York Times of ten years ago would probably stimulate my curiosity just as much as reading that of today. In some ways, it may do so even more. I agree with this, but I also suspect that we humans have a somewhat irrational bias towards favoring the “new” information. (Personally, I enjoy reading archives and past news coverage, as well as books written in a different era, though I don’t do much of this. Reading their predictions and comparing them with what actually happened is both amusing and sobering).

The Internet has changed the dynamics of how we work by meeting both these kinds of needs, and allowing us to jump from one to the other. On the one hand, newspapers and periodicals seem to be in a suicidal race to put up more and more of their content on the Internet, creating a very very huge stream of new information. On the other hand, basic reference material on the Internet is also growing rapidly. The most high-profile example of this is Wikipedia, the free online encyclopedia, that I’ve talked about in the past. But there are other, more low-profile examples, too.

For instance, the online bookseller Amazon is just one among many freely available online book catalogues. Many other retailers as well as libraries now maintain extensive online catalogues. Sites such as How stuff works and WikiHow have a lot of basic useful to-do information. More importantly, many organizations, including corporations and non-profits, are putting their data and advice online in a fairly well-organized form. Consider, for instance, societies devoted to medical conditions in the United States, such as Alzheimer’s Association. Or consider the many governmental websites (such as in the United States, but now also increasingly in less developed countries such as India) containing forms and help information that one would earlier need to go to a government office to obtain. Or consider the large amount of mapping information available that allows people to plan trips and find locations.

Linking the two forms of exploration better

Note: I use the term “newspaper” here as a generic term that may refer to a newspaper, periodical, or an online-only news/news commentary service.

Newspapers fulfill our need for “new” information, for information about things around us that we didn’t know we were looking for. For many of us, most of the time, this itch for new information is enough; we like to read and then forget the information and not think more about it. But it often happens that a particular news story makes one curious about further information on the topic. For instance, reading a story about recent progress in reducing infant mortality in an African country led me to wonder: what has been the general progress on infant mortality over the past century? Which countries have led and which countries have been behind?

To meet this kind of need, the newspaper needs to do something more than put an online version of a print article — it needs to link to pages that give an overall, better flavor of the subject, which may include past newspaper coverage on the topic, links to (in this case) sites on health/medicine, sites on secular demographic trends, etc. For instance, the New York Times has a topic page on infant mortality, from which I could see all the past coverage the New York Times had on the subject, as well as a multimedia chart comparing infant mortality across different countries and over time. Ideally, there would be a lot more information on the topic, neatly organized in the topic page, but this is a start. Unfortunately, many of the articles that the New York Times lists on this page (such as this one) lack backward links to infant mortality. (Incidentally, according to this blog post, the New York Times plans to release, in usable form, the linked data cloud that it uses for topic tags.)

Unfortunately, building this kind of topic page structure is a daunting challenge for smaller news and news commentary organizations than the New York Times. Many, such as Newsweek, have settled on a somewhat intermediate solution — they have topic pages, but the explanation part at the top is simply pulled from Wikipedia (with attribution) and all pages referencing the article are listed in a sortable table. (See, for instance, the Newsweek topic page on health care costs). Other organizations, such as the The Huffington Post, have tried a different, more sophisticated approach. See, for instance, the Huffington Post pages on John McCain and on wikipedia. However, even the Huffington Post does not advertise this extensive use of tags as a way of searching. Many smaller news organizations simply rely on search rather than any (externally visible) tagging system, probably because their scale and scope makes investment in tagging systems not worthwhile.

One of the solutions for the problem of lack of scale (i.e., it is hard for small news organizations to create separate articles or a separate taxonomy) may be for multiple news organizations to pool together their taxonomies, and even to share topic pages and results. The Washington Post/Newsweek company, to take one example, has such publications as the Washington Post, Newsweek, Slate, Foreign Policy, and The Root. These already link to each other, but by having pooled-together topic pages, they can reap more benefits. Small news organizations that do not want to invest in taxonomies can use public taxonomies, or those contributed by larger news organizations (for instance, that of the NYT).

Topic pages versus search

Some people have argued that powerful search features have largely made things such as tagging, categorization, and other forms of arrangement largely redundant. There is some truth to this argument. To quite an extent, the utility of categorization was to speed up extremely slow and cumbersome search processes, and the existence of quick computer-enabled search reduces that need.

But a close look would reveal that tagging today is a lot more powerful and useful than it could have been in a pre-digital era. Now, because of the lack of constraints of physical placement, the same topic can receive numerous tags. These tags allow a much larger number of cross-connections to be discovered. So, while the need for unsophisticated users to use tags/categories may have considerably reduced, the scope for their use by even marginally interested and sophisticated people has expanded considerably.

But the more important point is that a central hub, such as a topic page, is more than a mere tag. It is an intelligent starting point because of the way it collates together diverse kinds of information related to a topic. For instance, consider the New York Times topic page on Iran. This is more than just a list of pages referencing Iran, and its value lies in the way it gives a clear idea of the way the newspaper covers Iran.

Genuine advantage

News organizations in the United States are currently (As of 2009) experiencing severe financial problems, and have often been criticized of not innovating and adapting to the Internet culture. But even a little sampling of newspaper websites such as the New York Times, Time, Newsweek, the Wall Street Journal, and the Chronicle of Higher Education shows that news organization websites have been very innovative in adapting to the Internet to improve the quality of presentation of news even as they have to make painful cuts to their news staff.

In the past, newspapers were a one-stop shop for all kinds of information such as weather information (now, weather services across the world have websites and can send alerts to mobile phones), stock price information (again, stock exchange movements are now publicly available on their own and many other websites), and real estate and matrimonial information (that got them a lot of their ad revenues). In practically all these respects, there will probably never be a single one-stop shop.

There are two things news organizations should continue to be good at, and should lead in almost by definition: gathering news on-the-ground (thus meeting people’s instant-news-need) and providing a framework linking current news to past coverage (thus providing a vast reference corpus for answering specific questions people have, particularly those raised by current events). It is in the second respect that the shift to online news really helps improve news quality. But for this, there needs to be an effective mechanism to link the dots. I assume that part of a journalist’s training involves linking the dots and placing the news being gathered with their knowledge of history and past events. By making these links more public and sharing them with readers, news organizations can significantly improve the news value of their content.

As more and more amateur news gatherers come on to the scene and break a lot of news, even the ability of news organizations to manage to always be the first with new news may be in question (though newspapers are currently far ahead). Their large corpus of well-organized internal news archives can, however, give these organizations one advantage that will be harder to eliminate: the ability to put new stories in a broader perspective and grow topic pages and central hubs to provide readers with a quick overview of practically any topic.

Addenda

Concerns have been raised by many that the New York Times does not link to source content even when such content exists on the web, and that a large fraction of its links are internal links to Times Topic pages. Some suspect that it is a strategy to maximize PageRank juice. This is definitely a valid concern, though I think that of late, the Times has started to link more to outside sources as well, and it definitely does freely link for blog entries, and at least it links to websites when talking about content specific to those websites. In any case, whether the Times is suppressing outward links at a cost to readers is largely tangential to the question of the utility of Times Topic pages. Here is a book chapter about linking practices.

Million words on the first-year: Mocumentary of first-year life at Chicago

It is said that a picture is worth a thousand words. A six-minute video is probably a thousand pictures.

There is a tradition in the mathematics department of the University of Chicago whereby, at the end of each academic year, the second-year class organizes an evening of skits called “Beer Skits” (“beer” because the skits are accompanied by huge servings of beer). As part of the tradition, we decided this year to create a mocumentary (mock documentary) of life in the first year at Chicago. I had the original concept and fleshed out the main scenes, but a lot of the editing work before, during, and after the shooting was done by many of my batchmates, who all added their own insights and removed some of my original ones that would have been too abstruse.

Our six-minute video is a little funny, but it is also quite realistic, as those who have gone through a similar experience can attest. Okay, agreed, some of the scenes towards the end stretch the boundaries of realism, but almost everything we have is a slight adaptation of something that actually happened during our first year.

Below is an embedded Youtube video (you can make it full-screen for better viewing).

Download a high-resolution 3mbps DVD quality version in Windows Media Video format (104 MB).

Extensible automorphisms problem solved long ago

Some friends as well as other regular readers of this blog may remember that I mentioned the extensible automorphisms problem: if an automorphism of a group can be extended to any group containing it, must that automorphism be inner?

I had managed to prove this for finite groups, and had drafted a short paper with that and some other related results. My advisor George Glauberman then shared the results of this paper with some other mathematicians, and one of these people, Avinoam Mann, recalled that these problems had been the subject of papers in 1987 and 1990. You can see the extensible implies inner page on the groupprops wiki for more details.

Since I came up with the question myself, and also came up with the final answer myself (of course, with help and guidance from many others), I actually feel vindicated rather than distressed that this question was considered and solved long ago — it shows that the questions I ask have been asked by other people before, and been considered worth publishing.

In mathematics, one may be tempted to feel that it is getting harder and harder to get original results because more and more of the simple stuff has been taken. At the same time, as we acquire more and more knowledge, new researchers can stand on the shoulders of their predecessors and therefore ask and answer deeper questions. I feel that the benefits of being able to look farther and deeper outweigh the costs of having all the more basic results “already taken”.

I still do have a few additional results that I believe to be original, such as this one and its corollary, this one. I also have some partial results on the NPC conjecture, that I believe are new, such as this and others discussed on the NPC conjecture page. I hope to put these together with other results that I am yet to discover and hobble together something publishable in the not-too-distant future.

Tricki goes live

A short while ago, I blogged about the Tricki, mentioning what I considered an unnecessary subservience to Wikipedia. At the time, Tricki was still pre-live. Recently, Tim Gowers announced that the Tricki is “fully live”, which means that anybody can create a login and add and edit entries.

The Tricki already has a reasonable number of decent articles — an interesting article that I located in group theory is: first pretend that a normal subgroup is trivial. Here’s a link to the list of group theory-related tricki articles. Here is a list of tags that can be used to navigate the tricki.

The concept of the tricki is interesting, and I suspect that a lot of interesting stuff will go on there in the next few months. On the other hand, the main problem I see with the Tricki is that, while it has a lot of useful tricks, there is no easy standard reference it can use for looking up more details of the terms and facts used. The current consensus seems to be to link to Wikipedia, but using only one resource, particularly a resource of dubious value, seems restrictive, particularly since the specific Wikipedia article pointed to may not be particularly good. Consider, for instance, this page where, in the comments, Gowers notes that he “wouldn’t mind being reminded” what a characteristic subgroup is.

Here is what I suggest. For each of the concepts, Tricki should have a short page giving the definition and useful links, and the links from the tricki article pages should point to this short page. This way, article writers can concentrate on writing their articles rather than on defining all the auxiliary concepts involved (particularly in cases where these auxiliary concepts are reasonably standard and known to many of the readers of the article).

The concept page could serve three roles:

• A concise definition.

• A list of related tricki articles (this could be automatically generated by backlinks, or manually created and organized).

• A list of references/weblinks for more details. This list should be carefully curated, and people making such links should actually read the reference being linked to! Wikipedia could, of course, be one of the references typically given, but it need not be the only one.

Why do I think that separate concept pages are important? To take a totally different example, consider newspaper and magazine websites. Many of these websites have their own reference pages on a number of important topics — these reference pages provide a short summary, external links, as well as details of past coverage of the topic. For instance, The New York Times has Times Topic pages on almost all the things it reports on regularly, and its news articles have internal links to these Times Topic pages. While this is partly a tactic to maintain link juice, it also helps provide entry points for people seeking to get information on topics and get an overview of how the NYT has covered the topic. See, for instance, the Times Topic pages on Wikipedia, Twitter, and The Amazon Kindle. It seems to me that despite the presence of Wikipedia pages that may carry a lot more information, these Times Topic pages serve a useful function to readers who are interested in newspaper coverage of the material.

Also, there is something — a lot, in fact — to be said for keeping links largely internal, particularly when building up an extensively cross-referenced body of knowledge, information or insight. Part of Wikipedia’s success can be attributed to its policy of strong internal linking — extensive linking to other Wikipedia pages from within Wikipedia articles.

UPDATE: Gowers has started a forum discussion on some of the issues raised in this blog post.