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!

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.

There will be a workshop on F1-geometry in Granada, 23-25 Nov. This has not been advertised much before, so spread the word. They plan a number of introductory lectures on various aspects of geometry over the field with one element, and it all looks very exciting. As an extra bonus, you might also get the chance to see the legendary range of symmetry groups in the Alhambra.

A few more conferences have been added to the Events page. Some highlights for the coming months:

• Workshop on motives in Tokyo
• Advanced school and conference on homological and geometrical methods in representation theory, Trieste
• Arithmetic of fundamental groups, Heidelberg
• Algebraic Geometry and Arithmetic, Essen
• International School on Geometry and Physics. Geometric Langlands and Gauge Theory, Barcelona

See the Events page for dates and much more.

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.

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!

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.

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.

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.

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?