If you want to go beyond the Princeton Companion to Mathematics then the Oxford User’s Guide to Mathematics could be an answer

The Princeton Companion to Mathematics was extensively reviewed, and often praised, all over the mathematical and scientific blogosphere, see e.g. here, here and here. Most of this praise is probably well deserved. But where should an interested student (or even a professional mathematician who wants to extend her or his professional range, for that matter) go in order to deepen the knowledge acquired from PCM without getting bogged down into the details of the proofs and other such subtleties that abound in the specialized literature?

Of course, there is plenty of possible answers to this one, and you are welcome to share yours in the comments. However as far as “classical” (basically more or less up to the early XXth century level) mathematics goes, the Oxford User’s Guide to Mathematics appears to provide, at least for me, a reasonable, if not quite perfect,  enhancement for PCM.

OUGM has many omissions of its own and certainly could use more editing and proofreading — in particular, in order to make it somewhat more self-contained, but nevertheless this book provides a fairly broad and reasonably deep (for the beginner) panorama  of the “classical” mathematics as defined above. For instance, it does not cover category theory and related stuff. However, by and large, OUGM does a quite decent job in helping the beginner to advance her/his understanding of a great number of mathematical disciplines from abstract algebra to probability theory, and I certainly recommend to have a serious look into this book if you really want to deepen your knowledge of the “classical” subjects beyond the PCM level.

P.S. I just cannot miss this opportunity to wish merry Christmas and happy New year to the readers of this blog

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Recently updated posts

As I often update old posts instead of writing new ones , below is the (possibly incomplete) list of most recent updates:

• How to Write Grant Proposals (last update:  December 26, 2009)
• How to Choose a Research Topic (last update:  November 13, 2009)
• Career advice from the Fields medalists and some other mathematicians (last update: November 11, 2009)
• How to Write a Really Good Research Paper (last update: October 29, 2009)
• Hunting Down the Old References (last update: June 22, 2009)
• Academic Networking: on the Network and beyond (last update: June 22, 2009)
• Academia or Industry? (last update: June 6, 2009)
• How to Maximize Citations (last update: June 1, 2009)

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How to Write Grant Proposals

Grantsmanship Revisited

An important addition to my earlier post on writing grant applications: there is a series of articles on science funding in Science Careers. The author is someone going — quite appropriately –  by the name of  Grant Doctor.

Top Ten Open Problems in Physics

A list by Dmitry Podolsky

How Successful Mathematicians Work

I have found (hat tip: Yuri Kryakin)  a great interview in Russian with Ivan Panin, where he reminesces, inter alia, about his teacher, a prominent mathematician Andrei Suslin and the way he works.  The whole text is pretty long and very interesting but it is quite difficult to find reasonably self-contained excerpts to translate into English for those who don’t speak Russian, so let me give you just one bit as a teaser:

Suslin tackled the problems roughly as follows: first we see [the problem or the result to prove -- S.R.], then we believe [that we can solve it or that we can prove the result -- S.R.], and then we prove it. Because if you don’t believe, you will not have your vision materialized.

 

The Most Important Quality for a Working Mathematician

According to Sir Michael Atiyah, the most important quality for a working mathematician is the ability to maintain concentration for a long time.  Via Konstantin Zuev (original post in Russian).

Israel Gelfand (1913-2009). R.I.P.

Israel Gelfand (September 2, 1913 — October 5, 2009). R.I.P.

To learn more about him, see e.g. the excellent post by Terence Tao.

Hunting Down the Old References

While writing the research papers one quite often needs to get back to the full texts of old (pre-Internet or at least pre-arXiv) references. Of course, having access to a good library and/or the interlibrary loan usually solves the problem but can be somewhat time- and cost-consuming.

It is not that well known, however, that there is a fair chance to find the old paper or preprint you need online for free. Of course, the first thing to try is Google or perhaps another search engine of your choosing. However, if this does not work, you still have a fighting chance, at least as far physics and mathematics are concerned. The places to try are:

• the KISS preprint server (you can also try the umbrella interface at SPIRES)  allows you to search in (and get to the full text of) a huge database of scanned preprints  going back to the 1970s at least. The database covers mostly high-energy physics and related areas, including a fair share of mathematical physics and mathematics. For instance, you can find there a number of preprints by Richard Feynman, including the unpublished ones.
• the Digital Mathematics Library
• NUMDAM and CEDRAM (French mathematical journals)
• The Project Euclid
• MathNet.Ru (Russian mathematical journals)

All items but KISS are purely mathematical databases (to be precise, MathNet.Ru includes several physics, mechanics and mathematical physics journals as well).

If you know of other similar databases (be it in physics, mathematics, life sciences,…), please feel free to drop a comment with the relevant link(s).

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N. David Mermin: Writing Physics

This great lecture by a prominent physicist can be found here (hat tip: Biocurious)

How Much Passion Do You Need to Succeed in Science?

This very interesting and very controversial issue is discussed here, here and here. The discussion was triggered by this post at the YFS blog on the all not-too-nice kinds of people one encounters in science and on losing one’s illusions down the road into the academe (see also here for a related post at the RS blog); for a kind of alternative point of view see here.