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Some time ago the journal "Algebra and Analysis" (English translation is published in "St. Petersburg Mathematical Journal") had a special section which was called "easy readings for professional mathematicians", which tried to present in accessible and interesting way some ideas. It seems now this section is discontinued unfortunately.

So my question is the following: are there some journals which publish "easy readings"? Or may be there are some collections somewhere else on web?

PS May be some recommendations for concrete "easy reading" papers are also valuable.

PSPS As a kind of bigger and vaguer ingredient of the question, let me add the following: do you feel some over-representation of some hard-to-read papers over some beautiful and short papers? If mathematics is "mainly about follow-ups" it seems it makes progress more difficult, can something be changed or nothing ? One of reasons is probably the gap between "thinking and explaining", however as Gil Kalai answered he does not feel much gap, in part due to "Part of the reasons is being involved in semi-formal Internet acticities like blogs, polymath projects, MO, etc., ", this I think gives some hope that giving a bigger weight to user-friendly ways of communication may improve readability of papers...

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    $\begingroup$ Not quite a duplicate, but similar to this earlier question: mathoverflow.net/questions/15366/… $\endgroup$
    – Mark Meckes
    Commented Jun 13, 2012 at 10:46
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    $\begingroup$ @Mark thanks for reference, however seems to me many "expository" papers, are quite far from "easy readings"... $\endgroup$
    – Alexander Chervov
    Commented Jun 13, 2012 at 11:07
  • $\begingroup$ This List of Elementary Mathematics Journals at the MathPro Press webpages is at the low end of what you want. Not all their links are currently valid, however. For example, here's a correct link to The Pentagon. $\endgroup$
    – Dave L Renfro
    Commented Jun 13, 2012 at 14:52
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    $\begingroup$ I think that certain blog posts would fit your description of text well. A nice overview of math blogs is to be found here mathblogging.org ; in particular check the 'weekly picks'. $\endgroup$
    – user9072
    Commented Jun 13, 2012 at 16:16
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    $\begingroup$ Anton is correct, easy reading section of St. Petersburg Mathematical Journal still exists. As an editor, I would like to encourage the MO community members to apply. $\endgroup$
    – Fedor Petrov
    Commented Oct 25, 2017 at 5:12

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There are (at least) two journals which aim to publish that kind of papers: "L'Enseignement Mathématique" (http://www.unige.ch/math/EnsMath/) and "Expositiones Mathematicae" (http://www.elsevier.com/journals/expositiones-mathematicae/0723-0869)

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I am surprised that no one has mentioned the expository articles in the Bulletin of the AMS, which are usually excellent.

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    $\begingroup$ They are indeed usually excellent, but not always "easy reading" IMO. $\endgroup$
    – Todd Trimble
    Commented Dec 4, 2012 at 15:14
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    $\begingroup$ Agreed, Todd, and I'd say something similar about the "What is...?" feature in the Notices. It's a fantastic idea, and sometimes the articles are very good, but too often they assume too much specialist knowledge on the part of the reader. $\endgroup$
    – Tom Leinster
    Commented Dec 4, 2012 at 18:23
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There's a series of books, The Best Writing on Mathematics 201x, where currently $x$ runs from 0 to 6, edited by Mircea Pitici.

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While it's been some years now since I've read it regularly, my go-to for high-level "casual" mathematics writing has been The Mathematical Intelligencer, which has consistently done an excellent job of bringing high-level mathematics (e.g., de Branges's proof of the Bierbach conjecture) to mathematically literate readers in a literate, conversational style.

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SIAM Review has such papers every issue. From their website

"The SIAM Review consists of five sections, all containing articles of broad interest. Survey and Review features papers with a deliberately integrative and up-to-date perspective on a major topic in applied or computational mathematics or scientific computing. Research Spotlights publishes research or software papers in applied and computational mathematics that will appeal to SIAM Review's diverse audience. An article could take a non-traditional form such as a mini-survey or timely communication. SIGEST highlights a recent paper from one of SIAM's specialized research journals, chosen on the basis of exceptional interest to the entire SIAM community and revised and condensed as needed for greater accessibility. Education consists primarily of individual modules that are self-contained presentations of specific topics in applied mathematics, scientific computation, or their applications; each module provides the primary material needed to teach a given topic as well as supplementary material. "

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One enjoyable math article I read recently was Erica Klarreich's exposition of the virtual fibering conjecture. I would highly recommend this and other articles at this link to the Simons Foundation website. They use great diagrams and good analogies without shying away from difficult subjects.

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The Paul R. Halmos-Lester R. Ford Awards recognize authors of articles of expository excellence published in The American Mathematical Monthly. Most of these articles are available online (there is a 3 years lag) and are often "easy reading".

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There are prizes for mathematical exposition, scoped to a single work, which I think provide possible sources of easily-readable material.

For example:

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The Newsletter of the EMS (as well as the Gazette des mathématiciens, usually in French) often contains good exposition. The EMS also recently launched a journal for survey articles (but not always as broadly readable than your target, I think).

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