Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Archive for the ‘Uncategorized’ Category

Essay on Tate’s thesis

Posted by Andreas Holmstrom on June 14, 2010

Many years ago I wrote an essay on Tate’s thesis, which is now (finally) available here. This is the “baby case” of the global Langlands correspondence, and involves lots of interesting mathematics. Obviously there are many other introductions to Tate’s thesis on the web, for example this discussion on the blog of Terry Tao.

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Happy New Year!

Posted by Andreas Holmstrom on January 16, 2010

A bit late, I know, but I would like to wish all blog readers a peaceful and fulfilling new year 2010! Due to holidays, job applications, and a stolen laptop, I have been rather silent for a while, but hopefully I will be able to post more often in the next few months. Today I don’t have much to say, except showing you a picture from the year that passed:

Academy of Motives

Academy of Motives, Granada

It’s a tiny picture taken with a bad mobile phone camera, but it shows the Academy of Motives in Granada, where I was attending the F1 workshop in November. Other mathematical-touristic experiences of 2009 included two trips to Germany; unfortunately I was not able to find the famous shop where they sell affine schemes.

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Ordinary vs generalized cohomology theories

Posted by Andreas Holmstrom on December 10, 2009

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Blog silence because of Math Overflow…

Posted by Andreas Holmstrom on November 20, 2009

The last few weeks have been quite busy, and the spare moments that I would normally spend on blogging have been hi-jacked by Math Overflow. I wrote a few things there which I would normally have put on this blog, and since they might possibly be of interest to some blog readers, here are the links: Why are functional equations important, and What is the Yoga of Motives.

For quite a while, I have been trying (without much success) to understand finiteness properties for simplicial sheaves, and thanks to MO, I got an absolutely brilliant explanation from Denis-Charles Cisinski – something which simply could not have happened otherwise.  Lots of credit to MO (and to Cisinski)!

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Intersection theory at Rigorous Trivialities

Posted by Andreas Holmstrom on November 3, 2009

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!

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Semi-abelian categories

Posted by Andreas Holmstrom on November 1, 2009

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.

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Notes on p-adic Hodge theory

Posted by Andreas Holmstrom on October 15, 2009

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.

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Math Overflow!!

Posted by Andreas Holmstrom on October 15, 2009

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!

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Young Researchers in Mathematics conference in Cambridge

Posted by Andreas Holmstrom on October 15, 2009

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.

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Mathematical mailing lists

Posted by Andreas Holmstrom on October 9, 2009

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:

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

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 TheoryLondon 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.

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