Intersection theory at Rigorous Trivialities

Charles Siegel at Rigorous Trivialities is aiming to blog about intersection theory every day of November, essentially creating a minor book in the process. The first two posts are out, one on Chow groups and one on Manipulating cycles. These posts look promising, and I am very much looking forward to the rest of the series!

Semi-abelian categories

The usual setting for doing homological algebra is abelian categories. However, many of the things one can do in abelian categories also make sense in more general settings. For example, the category of groups is not abelian, but one can still make sense of exact sequences, diagram lemmas, and so on.

A more general framework for doing homological algebra, which I first learnt about from Julia Goedecke, is given by the notion of semi-abelian categories. Some examples of semi-abelian categories are: groups, compact Hausdorff spaces, crossed modules, Lie algebras, any abelian category, and any category of algebras over a reduced operad (although I am not sure what it means for an operad to be reduced).

A very nice introduction and survey of semi-abelian categories can be found in the recent article of Hartl and Loiseau, on the arXiv. Other references include the nLab page and the thesis of Van der Linden.

The category of monoids is unfortunately not semi-abelian, but there was an interesting discussion on Math Overflow recently about making sense of homological algebra in the category of commutative monoids, which is interesting when trying to do algebraic geometry over the field with one element.

Notes on p-adic Hodge theory

For a long time I have been looking for a sensible introduction to p-adic Hodge theory, and I think I might finally have found one: these lecture notes of Conrad and Brinon, an expanded but still prelimary set of notes based on their CMI summer school lectures earlier this year. Thanks to David Brown for pointing out these notes on Math Overflow, as part of an answer to a question about models.

A much shorter survey is Berger: An introduction to the theory of p-adic representations, but Conrad and Brinon give a lot more background, which seems very helpful.

Math Overflow!!

An amazing new questions-and-answers site has been launched, and I believe it will be a huge success! I asked in a recent post for a place to post algebraic geometry questions, and now there is a wonderful place for this (and other mathematical questions as well). Check out the SBS blog post and the site itself!

Young Researchers in Mathematics conference in Cambridge

Registration is now open for the next Young Researchers in Mathematics conference in Cambridge, which will take place 25-27 March 2010. See the conference webpage for more information.

Mathematical mailing lists

Lots of jobs, grants, conferences etc are advertised on mathematical mailing lists. I have never seen any good page on how you find these mailing lists, so I will try to list the ones I know about, and please add a comment if you know of others. If your mathematical interests are completely disjoint from mine, or if you are not interested in research mathematics at all, then maybe you should not read this post but check out this page instead.

The lists tend to be quite different in nature. Some (like COW) are relevant only for a specific geographic region, while others are more global. Some (like ALGTOP) seem to welcome all kinds of questions as long as they are well-informed and research-related, while others (like EAGER-GEN) seem to be more restrictive in what they allow. Some (like the arXiv lists) come as RSS feeds if you prefer that.

There are some lists that should exist but do not, as far as I’m aware. One thing I really miss is a list for algebraic geometry which allows for all kinds of (intelligent) questions, in the ALGTOP style. Maybe algebraic geometry is too big a subject for such a list, but there certainly could be lists for arithmetic geometry and maybe also homotopical/derived algebraic geometry, and lots of other algebraic geometry subfields.

My favourite subject-specific lists are:

• ALGTOP (Algebraic topology)
• K-theory archive
• Categories list
• EAGER-GEN (European algebraic geometry)

When doing some googling for this blog post, I also found the following:

• Geometry

which I have now subscribed to.

A very useful thing is the arXiv mailing list, where you can specify what subject categories you are interested in. I have been subscribing to this for a while, but it’s hard to keep up to date with the emails, especially if you are interested in many subject areas. Am now trying the RSS feeds instead in Google Reader, one advantage being that it is easier to quickly skim through large amounts of posts. The only disadvantage is that I haven’t figured out how to eliminate duplicate feed items, which occur when a preprint is listed in more than one subject category, but I am sure there must be a clever way of resolving this.

A very general list is sci.math.research, where you can ask almost any question and usually get a sensible answer.

Some lists which are relevant if you are based in the UK: London Number Theory, London Geometry and Topology, and COW (see also the COW web page if you don’t know what COW is). When searching for mailing lists on various topics I also found the Midwest Topology list, which might be of interest to some.

Many research institutes have their own mailing lists, for example the Fields Institute, MSRI, and the Newton Institute. See this list of research institutes for more.

There might also be mailing lists from sites advertising math-related jobs, such as mathjobs and jobs.ac.uk, but I plan to come back to this and other jobs-related resources in a later blog post after doing some proper searching.

Simplicial homotopy theory book

Maybe someone will be happy to learn that the book Simplicial homotopy theory by Jardine and Goerss is now available in a softcover edition. The new edition is more reasonably priced, for example £36 on Amazon UK. This is around half the price of the old one, which was part of Birkhäuser’s super-expensive “Profit in Mathematics” series (sorry, “Progress in Mathematics” series). The book is excellent, being one of the few places where one can learn about many fundamental notions of abstract homotopy theory.

See the Google Book page for more info about the book, and a preview of a certain number of pages.

Toen on homotopy types of algebraic varieties

Two recent conversations both reminded me of a short note of Toen, with the title Homotopy types of algebraic varieties. This note explains in only eight pages several exciting ideas, which I find interesting especially because they point towards some possible future interactions between homotopy theory and arithmetic geometry.

He starts out by a conceptual discussion of classical Weil cohomology theories, which were discussed in this earlier post. The idea is that the cohomological invariants should be refined into some notion of “homotopy type”, the relation being somewhat analogous to the relation in algebraic topology, between the cohomology and the homotopy type of, say, a CW complex. He then goes on to sketch how this can be made precise, using the language of stacks and schematic homotopy types.

Towards the end of the paper, he speculates about a possible connection between the homotopy types of a variety and rational points on the variety. The study of rational points is one of the main themes of arithmetic geometry, as they correspond to integer or rational solutions of (systems of) polynomial equations. The famous section conjecture of Grothendieck, explained in these notes of Kim, is supposed to give a conceptual proof of Faltings’ theorem, aka the Mordell conjecture. Faltings’ theorem says that a curve of genus at least 2, defined over , only has a finite number of rational points. Toen suggests a generalization of the section conjecture to higher-dimensional varieties, using his notion of homotopy types.

Another main theme of arithmetic geometry is L-functions of various kinds. To any variety over , one can attach an L-function, which encodes lots of information about the arithmetic properties of the variety. Many outstanding conjectures in number theory are formulated in terms of these functions, for example the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, as well as many other more accessible conjectures. The building blocks of an L-function are precisely the various Weil cohomology groups, and one could speculate about the significance of Toen’s conceptual approach to Weil cohomologies. Could it give us some new tools for approaching questions about L-functions? Or could it be that L-functions are not the right thing to consider, but that the notion of homotopy types could lead us to some better objects of study?

Events in the coming year

Over the past few weeks I have been updating the Events page, and there is now a long list of interesting conferences, workshops and summer schools taking place in the coming year. I wish there was a more sensible way of finding and listing events, like a “conference section” on the arXiv, following the subject classification used for preprints. The current way of doing things, with random mathematicians posting random lists of events on various webpages seems very inefficient for everyone involved. Anyway, if you share some of my interests then you are likely to find something useful in the updated list. Note that if you are young there is often funding available from the conference organisers if you apply early.

Some highlights:

• A workshop in Leiden in October, under the somewhat obscure name Frobenius lifts. One of the main topics covered is so called Lambda-algebraic geometry, which is a new kind of geometry related to the field with one element. See this preprint of Borger for more exciting details!
• A one-year research programme in Barcelona on arithmetic geometry, with a number of workshops.
• An Advanced School on Homological and Geometrical Methods in Representation Theory, in Italy in January/February.

See the full list for much more…

Introduction to journals

Most of the output from mathematical research ends up in a mathematical journal. However, there is a multitude of such journals, and it can be hard to know which ones are most useful. In between my attempts to write a PhD thesis, I have been trying to get some idea of how maths journals work, which ones are good, which ones you can access online etc, and this process has now resulted in a resource page on maths journals. First of all, there is a list of around 45 journals which are my personal favourites, and I think this should give a good idea of which journals are most interesting, if you share my mathematical interests (arithmetic and algebraic geometry, algebraic topology, K-theory, category theory, and in particular cohomological and homotopical methods in these fields). I have included nonspecialized journals such as Proceedings of the AMS and Duke Mathematical Journal, whenever I feel that they tend to publish a significant number of articles I find exciting, and I believe I have included all high-quality journals specializing in the subfields just mentioned. If your interests are only partially overlapping with mine, or if you want to explore journal beyond the ones I have listed, it should be fairly easy to make your own list of favourite journals. Just throw out the irrelevant specialized journals from my list, and have a quick browse through the journal sections of the major journal publishers/providers – there are direct links to more or less all of them at the bottom of the resource page. Most journals have fairly informative names, although the Topoi journal from Springer did not quite live up to what the name seemed to promise.

All in all, there seems to be somewhere between 500 and 3000 mathematical journals, depending on what you count as a journal, and depending on what you mean by mathematical. Some of these are freely accessible to everyone, and for some you need to either subscribe or pay for each article individually. Many journals have a moving wall, meaning that you need to pay in order to access the most recent volumes (typically the last 5 years), while earlier volumes are free. If you want to know when a new issue is published, most journals have email alert services for this purpose.

By now most journals are completely digitized and available online, provided you (or rather your university library) have the right subscription. Of the journals in my list, I think the only exceptions are Journal of Algebraic Geometry, Comptes Rendus, and Memoirs of the AMS. The Memoirs will become available from early 2010 according to the AMS web page, although it is not clear whether this will include only new volumes or all previous volumes as well. The Journal of Algebraic Geometry volumes are available online starting from 2002, but the AMS Customer Service is not aware of any plans to digitize older volumes. As for Comptes Rendus, ScienceDirect provides access back to 1997, and hopefully earlier volumes will become available eventually. When it comes to book series which to some extent resemble journals, the Lecture Notes in Mathematics recently became available through SpringerLink, which is terrific, provided your university has a subscription. The early issues of the Bourbaki seminar are supposed to come online for free during 2010 at NUMDAM, but for the many wonderful Astérisque volumes, there are only tables of contents online, and as far as I know no plans for digitizing in the near future.

Many journals are available from more than one source, and sometimes one source is free while the other is restricted-access and very expensive. A notable example is Inventiones volumes from before 1996, which are available from Springer through subscription or by paying $34 per article, but are also available for free at the DigiZeitschriften site. The AMS Digital Mathematics Registry has a quite exhaustive list of journals where you can see all online sources for any given journal, which is sometimes very convenient.

There is a lot one could say about journal pricing, the serials crisis, the open access movement, and the evil deeds of certain publishers such as selling complete nonsense or hidden advertising under the cloak of a scientific journal. However, I will only refer to the EUREKA journal watch site, which has lots of material and many excellent links. It is interesting to realize that even a big and relatively rich university such as Cambridge can only afford to provide very partial access to the online mathematical literature. For example, I cannot access major journals such as Proceedings of the London Mathematical Society, Duke Mathematical Journal, or Journal für die Reine und Angewandte Mathematik, unless I get access through some other university.

There are various so called metrics which are supposed to measure the quality of a journal, or the quality of an individual researcher’s output. Other people have already written about why the Science Citation Index and the Impact Factor are stupid and dangerous. There exist attempts at constructing better measures, for example the eigenfactor metric.

To find specific articles you are looking for, the most efficient tool appears to be Google Scholar. Although still in beta, it almost always gives you what you want if you search for the author name and some keywords from the article title, for example the phrase “absolute Hodge author:Beilinson“. It has also become quite good at finding direct links to freely available preprint versions of journal articles with restricted access, and it has links to Google Books, which means that you can sometimes get to read articles from various conference proceedings, which otherwise typically exist only in paper form.

To organise articles you find online, there are various pieces of software, such as Endnote, Zotero, and Mendeley. I have only tried Mendeley, and although the idea looks absolutely brilliant, it did not work very well. In particular, almost all the automatically retrieved metadata was completely wrong, expecially for articles in French and older articles. If anyone has positive experiences with Zotero or some other similar software I would be very interested.

Finally, I would like to recommend this incredibly well-written blog post by Michael Nielsen, discussing the future of scientific publishing.

There are also other links at the journal resource page. I wish you a good summer and happy journal browsing!