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How does one find out what's happening in contemporary mathematics research?

EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside academia for quite a few years, and the gap between grad students' access to recent research (which is in the air, word of mouth + all publications immediately available) and that of a graduate who now works in an industry (few contacts if any + no access to journals) is tremendous. IMO this gap is the primary reason it's so difficult to do mathematics "on the side".

There are a few personal blogs out there, but personal blogs, even as very good ones, cannot be all-encompassing and necessarily focus on the subfields that the blogger is interested in most. There is MO, of course, but this is a Q&A site rather than a newsroom. There is word-of-a-mouth from other mathematicians, but the breadth of news one learns that way depends too much on one's contacts.

Physicists have Physical Review Focus website, which provides very accessible overviews of both recent and significant developments in Physics. This is not somebody's personal blog, but a well-organized collection of short digests by many contributors with references to the original papers. On most of the recent significant developments one gets a high level overview written by a specialist for a non-specialist, and can get access to a more detailed paper by following the supplied references.

I wonder if there is a site like that for mathematicians. Please tell if you know of one. If not, perhaps one could be built based on MO resources?

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    $\begingroup$ I suggest that the current title of the post be replaced by the first sentence of the post. $\endgroup$
    – Yemon Choi
    Jun 20, 2014 at 2:06
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    $\begingroup$ Have you tried looking at the arxiv and journals? Generally speaking looking at published work is a good place to start. $\endgroup$
    – Ryan Budney
    Jun 20, 2014 at 2:14
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    $\begingroup$ @PedroLauridsenRibeiro, unfortunately, mathscinet is not exactly free. $\endgroup$
    – Michael
    Jun 20, 2014 at 3:50
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    $\begingroup$ This is a great question and I have voted to reopen. It is indeed a vital problem to know what is going on in research and since there is no easy answer, every bit of information (especially from top research mathematicians) is extremely precious. As a first step in the right direction, I would be very grateful to the closers if they told us a little about their own field of research and their contribution. $\endgroup$
    – Georges Elencwajg
    Jun 20, 2014 at 7:13
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    $\begingroup$ @AndyPutman your advice is good but your assertion seems too strong (even if you include equivalents of MathSciNet such as zbbMATH). AFAIK access to MathSciNet (or an equivalent) is not even universal throughout the professional community. Furthermore, since zbMATH is free for focused searches it, in combination with free resources such as Google Scholar, can be used as an incovenient but still rather feasible substitute, in my opinion. $\endgroup$
    – user9072
    Jun 21, 2014 at 9:24

9 Answers 9

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The question may be understood in different ways, depending on the time scale (what do you mean by "recent"? last week? last year? last three decades?) and the level at which you wish the mathematical developments to be presented (for the general public? for beginning graduate students? for professional mathematicians in general? for specialists of the field?). So the question will not have any single good answer.

Also, and here it is perhaps a difference with physics, short time scale are not very relevant in mathematics. Most of the important new ideas takes years, often decades, to be checked, developed, known, understood. So it is not a big deal if you miss the first time (or the second the third, etc.) that an important development is announced: you'll have years to catch up, and after sufficient time the news will come from so many directions that you will not be able to miss it. This may explain why there is no equivalent of the Physical Review Focus in mathematics.

That being said, let me discuss the resources I know. At a very short time scale, the newsletter Headline and Deadline of an AMS announces surprising advances (e.g. the bounded gap between primes). Mathoverflow, even though it is not its aim, also does in practice. On a longer time scale (about 5 or 10 years), one famous resource is the Bourbaki Seminar, which tries to cover important development at a specialist level. Of course, it has never been perfect in covering all mathematics, and my opinion is that it is much less good in that respect that it used to be. Yet it is a valuable resource.

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    $\begingroup$ I would have to disagree about timescale: there have been many important developments with relatively short windows of opportunity for important research. Sometimes getting to the subject just one year after it was introduced means you shouldn't bother, the subject is effectively closed. Thus announcements with short high level overview on timescale of a few weeks would be very important to those not in direct contact with the leading specialists. $\endgroup$
    – Michael
    Jun 20, 2014 at 3:58
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    $\begingroup$ Interesting. Can you give some examples of such "short windows of opportunity for important research"? I can perhaps think of a few examples in my own field, but then the only people who could possibly make use of that "short window of opportunity" are expert themselves, who'll get directly informed by their network (seminars, conferences, papers submitted on arxiv and to journals, friends) of the developments opening the "window of opportunity". And even then, research done through those windows will not be that important. $\endgroup$
    – Joël
    Jun 20, 2014 at 11:23
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    $\begingroup$ @Joël: Here are some very fast development examples, from 3 different decades: 1. Date-Jimbo-Kashiwara-Miwa approached in 1981 non-linear PDEs with affine algebra representations; subject effectively closed in 1982. 2. Seiberg-Witten invariants appeared in 1994; most striking applications, such as the proof Thom's conjecture, were completed within a year. 3. Strengthenings of Zhang's results by Tao & Co and Maynard came out within months from the Zhang's paper, sometimes even BEFORE the original paper was published... $\endgroup$
    – Michael
    Jun 20, 2014 at 15:01
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    $\begingroup$ connection with the field which comes from years of work in the subject. That's in my opinion the main problem for an outsider doing mathematics: when there is something to do fast, people with a lot of experience in the field will in general be way faster. But this problem does not only concerns outsider who has left mathematics. It also concerns in a fundamental way PhD students. A well-informed PhD student, which goes to many seminars and attends conferences and speaks to many people so that he hears of everything that's happening in his field is still at an enormous disadvantage to... $\endgroup$
    – Joël
    Jun 20, 2014 at 15:30
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    $\begingroup$ use with profit a new discovery, compared with an older practitioner of the field. This is in part compensated by the help of his PhD advisor, but a PhD students who likes to works alone and is independent-minded (certainly an important quality to do good research in the long run) will have this disadvantage, and the solution is to work on problems which are not very time-sensitive. Fortunately, the are the majority of interesting math questions. $\endgroup$
    – Joël
    Jun 20, 2014 at 15:32
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I'm envious of the physicists for having a website like the APS website. Unfortunately in math, I think expository articles often lag far behind the original publications.

Regardless, here are some sources I use.

  1. Sign up for the ArXiv daily mailings, being specific about the fields of your interest. Always read the abstracts (they're displayed right in the e-mail), and read the introductions of the papers that interest you. This is tough going because many papers are hard to read, but this is a pretty good way to stay up-to-date.

  2. Check out the ICM talks (at least their titles). Though these are not a complete overview of modern math, and though they only occur every four years, they'll give you enough math to think about for more than four years.

  3. If you know the field or topic of your interest, try contacting a researcher directly. I do that when I don't know about a certain field of math, and I've found we mathematicians are quite helpful to each other. If you write an e-mail explaining your background, I bet you will find a good number of mathematicians who are at least willing to e-mail or Skype with you. (Just don't begin your e-mail with "I have squared the circle and proved the rationality of pi.")

  4. Check out videos at MSRI's website (for math in general) and the Simons Center website (for more math-physics stuff). These videos are a great resource, and workshop videos especially contain introductory overviews of contemporary topics.

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The AMS series

What's Happening in the Mathematical Sciences, Volume 1—9

is superb, but perhaps a bit slow compared to a newspaper or blog.

          Volume9
Table of contents of Volume 9 (PDF download).

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    $\begingroup$ Looking at the Table of Contents of volume 9, it seems interesting indeed, but very oriented toward applications, and not at all representative of what's happening in pure mathematics. $\endgroup$
    – Joël
    Jun 20, 2014 at 1:33
  • $\begingroup$ @Joël Aren't you going to recommend a certain venerable French organization? $\endgroup$
    – Felipe Voloch
    Jun 20, 2014 at 1:38
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    $\begingroup$ Unfortunately, not everybody has access to AMS publications. Are there online equivalents accessible from outside academia? $\endgroup$
    – Michael
    Jun 20, 2014 at 4:03
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    $\begingroup$ On the other hand, "slow" compared to physics research is not really very slow. Math papers ten years old are still quite current, while physics papers ten years old are often obsolete. $\endgroup$
    – Gerald Edgar
    Jun 20, 2014 at 15:07
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    $\begingroup$ Unfortunately, not everybody has access to AMS publications. Spend a few dollars! Since you are employed in industry, you can afford it. Yes, they sell to non-mdembers. $\endgroup$
    – Gerald Edgar
    Jun 20, 2014 at 15:08
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There is the monthly journal Notices of the American Mathematical Society.

According to its site,

(...) by publishing high-level exposition, the Notices provides opportunities for mathematicians and students of mathematics to find out what is going on in the field. Each issue contains one or two such expository articles that describe current developments in mathematical research, written by professional mathematicians. The Notices also carries articles on the history of mathematics, mathematics education, and professional issues facing mathematicians, as well as reviews of books, plays, movies, and other artistic and cultural works involving mathematics. Members keep abreast of official AMS reports, activities, and actions, and the news of the mathematical world, through articles the Notices.

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    $\begingroup$ And other professional societies have similar journals, e.g., SIAM Review or DMV Jahresbericht (in German). Really, membership in a professional society is an excellent way of keeping in touch with the profession (after all, that's what they specialize in). (I know @Michael asked for free resources, but individual membership fees are really moderate -- and tax deductible. It's also worth checking out if your employer is a corporate member and can give you access.) $\endgroup$
    – Christian Clason
    Jun 21, 2014 at 17:42
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Some conferences may post lectures online. For example, the Banff International Research Station (https://www.birs.ca/) hosts a 5 day workshop nearly every week on a contemporary topic with leading researchers, and you can watch live on their website. If you missed one you wanted to see, they also maintain a video archive for many of their talks.

(edited to include suggestions from Geoff Robinson and Yemon Choi)

There are also research lectures available from

The first three can also be watched live.

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An excellent resource is the collection of the talks given at the congresses of the International Mathematical Union. These are available, in browsable form, online at www.mathunion.org/ICM from the first one (1893) to the last one in 2010.

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This doesn't directly answer your question, but some universities give guest access to visitors for limited periods of time (e.g. an hour a day), so if there are any resources you really want to check out like MathSciNet or some paper whose author has not provided a arXiv version, you can read it at such an institution.

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1) Research statements.

Search Google for "research statement" and an area of mathematics that you are interested in.

Here's a short selection:

http://people.uwec.edu/mbirika/Aba_Mbirika_Research_Statement.pdf

http://www-personal.umich.edu/~satriano/research.pdf

http://www.math.uconn.edu/~Ji%20LI/Research-statement-JiLi.pdf

2) Online availability of prefaces, contents, supplementary material, and sometimes draft versions for new books.

Use Google advanced book search to Search for all books published in the last year (or further back as you wish) and scroll through the listings until you find something that looks new to you. Once you find a promising looking book, after trying Google preview, go directly to the publisher's page for the book and also try to find the author's home page to look for additional material.

Example: At http://books.google.com/advanced_book_search enter "mathematics" in the subject box and enter "topology" in the "Find results" box. Publication date 2014-2014.

On the second page of results the book "Topological Signal Processing" looks new.

Publisher's page: http://www.springer.com/engineering/signals/book/978-3-642-36103-6

Author's page: http://www.drmichaelrobinson.net/research.html

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Check out the programs and workshops at IPAM, http://www.ipam.ucla.edu/. IPAM is particularly strong on cross-disciplinary applied topics.

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