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What are really helpful math resources out there on the web?

Please don't only post a link but a short description of what it does and why it is helpful.

Please only one resource per answer and let the votes decide which are the best!

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I edited your question a bit, my main goal was to remove the reference to WolframAlpha; and make the text easier to read. But feel free to revert if you like! (You can see the edit history and revert by clicking on "edited ... ago") –  Ilya Nikokoshev Nov 4 '09 at 21:23
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The answers below are great. Here's an idea I don't see that may be interesting to think about starting: An online example repository. This would be a place where one could upload and search various "first nontrivial example" notes. I think such a thing would really move mathematics along. –  Jon Bannon Apr 20 '12 at 13:38
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66 Answers 66

up vote 86 down vote accepted

http://www.wikipedia.org

I have learned a lot of mathematics while reading Wikipedia. Allowing a wide audience to contribute to articles seems to work out well in the case of mathematics.

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See especially the mathematics portal en.wikipedia.org/wiki/Portal:Mathematics on which page there are links to more-specific mathematical portals, and to mathematics categories. –  Rhubbarb Nov 11 '09 at 9:16
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I occasionally find mathoverflow.net rather helpful.

In particular, there's a good list of answers to your specific question here.

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interesting occurrence of the Droste effect, en.wikipedia.org/wiki/Droste_effect –  Sean Tilson Apr 18 '10 at 2:58
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@Cam, you've taken MO off the list of helpful math resources that don't list themselves, thereby averting Russelian calamity. Good on ya. –  Gerry Myerson Nov 16 '10 at 3:13
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mathoverflow.net? Never heard of it. –  NPC Nov 16 '11 at 8:40
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I don't like this website. It advertises itself in one of its threads. –  Niemi Aug 28 '12 at 8:36
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I use http://arxiv.org/ all the time.

Researchers post their articles here, so it is a great way to see if anyone have already a proof or an idea on something. Some people regularly access it through

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and they have rss feeds! –  Andreas Holmstrom Oct 25 '09 at 4:56
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For enumerative combinatorics, it's hard to beat Sloane's Online Encyclopedia of Integer Sequences.

It is what it says on the tin. A huge list of integer sequences, with references, links, formulas, and comments.

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Of course there's much more than enumerative combinatorics -- it's very strong in number theory, for example. –  Charles Oct 7 '10 at 1:06
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Mathscinet, which contains summaries and reviews of published research papers. It's very useful when you want to get an idea of a paper without having to read it, and contains almost every paper ever published.

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I agree. But it's unfortunate that often the "summary and review" often consists of nothing more than the abstract of the article and the bibliography. –  Grétar Amazeen Oct 27 '09 at 17:23
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Of course every mathematician should be familiar with this. Be glad you are doing it today. Back in the day I would have to walk to the library, search through multiple index volumes of Math Reviews, and maybe find what I wanted. But I still did it once a week or so... –  Gerald Edgar Oct 28 '09 at 13:51
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Beyond the reviews at MathSciNet one of the most beautiful features of the website is that it allows you to page back and forward through the history of the topic at hand. You're reading the review and you can click on the link that offers you the papers that cite this paper. So if you happen across an old paper that's concerned with a topic you're interested in, and want to know if anyone has done anything more recently on that topic, you're just a click away. I don't know when that feature was added but when I was a young mathematician I would have killed for that feature. –  Ryan Budney Nov 5 '09 at 4:15
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I'm not in academia any more and so can't access Mathscinet (and JStor). It's very frustrating. –  Anonymous Mar 8 '10 at 16:39
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Finding forward and back references is indeed one of the most amazing and useful things about MathSciNet, but be aware that not everything will show up. Articles collected in books and older articles don't have their references in the database. The cutoff seems to be around 2000, but it may be different for different journals. –  Tom Braden Jun 18 '10 at 12:43
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Everything by John Baez. In particular This Week's Finds in Mathematical Physics, the n-Category Cafe, and the n-Lab. He has an amazing ability to make even the most esoteric topics seem obvious and inevitable.

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I'm partial to both John Baez and the nLab, but the two don't have much intersection any more. The nLab is sometimes short on making really esoteric topics seem obvious. –  Todd Trimble Aug 28 '12 at 11:06
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http://jmilne.org has lots of systematic, well-written courses.

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Terence Tao blog

contains a lot of useful advice for people at various stages in their careers. In addition it contains a lot of discussions and explanations of the math that I find interesting.

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Detexify is a quick and easy way to find the name of a LaTeX symbol.

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http://scholar.google.com/

Allows you to search for research articles. Gives you direct links to all online versions of the article it can find. Strenghts include that it can often give you direct links to files hidden on obscure (non-arXiv) preprint servers or personal webpages, and if you sit on a computer with access to ScienceDirect, Springerlink etc, you get direct links to these artices, via your university library. Weaknesses include lots of errors due to the reliance on their "intelligent" search engine rather than correct metadata from publishers, but this is likely to improve over time.

To find what you really are looking for use the author tag, for example "infinite loop author:May" etc.

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Don't forget http://gigapedia.com/ for tons of e-books.

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Several people have flagged this answer as requiring moderator attention. Please use your own judgement as to the legality of any site on the internet, and remember linking is always okay. –  Scott Morrison Mar 8 '10 at 2:24
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We have enough police in this world already. –  I. J. Kennedy Oct 27 '10 at 1:33
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To quote a lower post by Henning Arnór Úlfarsson, "Alexandre Stefanov keeps an extensive list of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there." (Link is mathbooks.110mb.com/mylist.php) This can work as a substitute for most of the titles one would want to find from gigapedia. –  drvitek Oct 27 '10 at 5:10
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Simon: try trillia.com/online-math/index.html for a modified version that somebody got. –  drvitek May 3 '11 at 15:49
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http://books.google.com/

If you haven't figured this out already, you can read large portions of textbooks before you buy them to decide if they're what you need. (If you actually want free books, there's a separate question that addresses that.)

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One extremely useful way to use Google Books is as an "instant index." For example, I have the book ZZZ open in front of me. There is a term yyy I don't know the definition of. I look up yyy in the index of ZZZ. Ooops, no index! So I search for yyy at the Google Books page for ZZZ and order by pages. Done. –  Sam Nead Mar 8 '10 at 14:24
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An annoying feature of this site is that a lot of the book portions do not include the preface or introduction which is usually the first thing I want to read about a book. –  Anonymous Mar 8 '10 at 16:41
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The preface/introduction parts can be often found in the book preview on amazon.com. For the books published by the AMS these are often availble right on the AMS bookstore website (ams.org/bookstore). –  mathphysicist Jun 4 '10 at 17:43
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The open source software package SAGE at sagemath.org can calculate, well, almost anything you want. The mission of the SAGE group is: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.

The most useful resource online is www.sagenb.org, where one can log in and use SAGE online, without having to install any software.

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The nLab is an excellent resource, often containing more detail, explanation, and discussion than wikipedia, along with much more specialized and contemporary topics.

(nLab was mentioned in the answer by Justin Hilburn, but it was listed there after other resources, and I think people scanning under the one-resource-per-answer dictum will miss it.)

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http://mathworld.wolfram.com/

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The book for this website is the "Concise Encyclopedia of Mathematics" by Eric W. Weisstein, CRC Press. The website is "Eric Weissteins's MathWorld", from which there are links to areas of mathematics and to other on-line mathematical tools. Both book and website take the form of a mathematics dictionary. This site is excellent, but beware that as it is associated with Wolfram Research, there is a bias towards the commercial mathematics package "Mathematica". –  Rhubbarb Nov 11 '09 at 9:26
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MathOnline

Recently launched by Andrea Ferretti

Here one can collect lecture notes, survey articles, books and so on. All the material can be organized and searched by author, topic, language, level and so on.

Registered users can add new books, add tags, write reviews, vote, keep a list of the favorite books and see other people's profiles.

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I am surprised nobody yet have put pointers to books and papers. For older stuff you can find a lot at Gottingen Digital Library, Numdam and JSTOR.

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Sloane's OEIS has already been mentioned.

A similarly useful site is ISC, Simon Plouffe's Inverse Symbolic Calculator.

Here you enter the decimal expansion of a number to as many places as you know, and the search engine makes suggestions of symbolic expressions that the expansion might be derived from. The answer might involve pi, e, sin, cosh, sqrt, ln, and so on.

Sometimes, it becomes difficult to calculate symbolically. Therefore, you can proceed numerically instead, and hope to recover the exact symbolic solution at the end, using ISC: sometimes proving that an answer is correct can be easier than calculating, or discovering, it in the first place.

It can also be useful for discovering simplifications of nested radicals, for example.

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The Tricki

Quoting the site:

"Welcome to a brand new Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques. Some of these techniques will be very general, while others will concern particular subareas of mathematics. All of them will be techniques that are used regularly by mathematical problem-solvers, at every level of experience."

http://www.tricki.org/

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Unfortunately Library Genesis is down and has been down for some time now, so I'm taking the liberty of editing this answer. A working site that is similarly useful is libgen.info and someone has collated a blog of links here.

Original answer follows:

At the Library Gensis or the translated version here, you can browse and download as many high-quality and modern Mathematics books, surveys, etc as you wish. This i-resource must be on every mathematician's i-shelf.

Here is some list:

599 books on Number Theory;

303 books on Complex Analysis;

325 books on Algebraic Geometry;

588 books on Partial Differential Equations;

97 books on Abstract Algebra;

107 books on Commutative Algebra;

181 books on Harmonic Analysis;

133 books on Fourier Analysis;

349 books on Functional Analysis;

356 books on Differential Geometry;

88 books on Riemannian Geometry;

783 books on Topology;

286 books on Combinatorics;

323 books on Graph Theory.

This is enough for illustration. You will find more, enough to get you in a downloading craze!

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The link, however, is one of the most useful locations of the internet. –  darij grinberg Apr 10 '11 at 15:22
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Aside from the copyright issues, I find this site and similar (extracoder, eknigu, lib.by etc.) one of the most useful math resources from the web. –  Marcin Kotowski Apr 10 '11 at 18:31
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Rather than 'aside from the copyright issues', I would say 'because of the copyright issues'. If we had a working system of copyright, we would not have to resort to questionably legal means of getting information that naturally belongs to all of humanity. –  A. Pascal May 29 '12 at 7:53
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http://eom.springer.de

Very good articles with lots of references! (never mind the .de, it's in English!)

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Free downloadable (and streaming) video lectures from the Institute for Advanced Study at Princeton: http://video.ias.edu/

Not exactly a resource, but a great way to listen to talks given by experts on the latest results in Computer Science and Math.

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The Art of Problem Solving

Mostly for the student, including high school. But has more advanced forums, too. Latex easily used.

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I learned more in high school from reading posts on Art of Problem Solving than from any other source. Of course, this is before I discovered the blogosphere. –  Qiaochu Yuan Nov 5 '09 at 3:39
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http://www.wolframalpha.com/

I'm just adding Wolfram Alpha to the fray so it can be voted on like other suggestions. For people who haven't heard of it, it's an online computational engine.

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Paul Stuart says: I think Wolfram Alpha is just a waste of computational effort. Whatever comes from the Wolfram corner I'd be wary. It ranges from acclaimed proofs to acclaimed "correct computation". Note they are infamous for making claims or assertions that aren't true. So, think twice. –  Anton Geraschenko Nov 6 '09 at 18:21
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Qiaochu Yuan says: In all honesty, I use Wolfram Alpha whenever I can't be bothered to remember what the real syntax for a command in Mathematica is. –  Anton Geraschenko Nov 6 '09 at 18:22
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dimitry (user 1087) says: @Qiaochu: That's just simply sad news altogether. There is a good point raised by Paul Stuart and that is the point raised about reliability. Wolfram Alpha, is a black box. God knows how reliable its answers are. If I aint completely sure, Wolfram had been (not unfairly) accused of being sloppy in many of his claims in his previous work on ANKS. If this is based on work in ANKS I'd be wary too. –  Anton Geraschenko Nov 6 '09 at 18:25
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Qiaochu Yuan says: Math Overflow is not a forum. It has no collective opinion on Wolfram, and this is not the place to discuss your opinion of his work. –  Anton Geraschenko Nov 6 '09 at 18:28
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http://wiki.henryfarrell.net/wiki/index.php/Mathematics/Statistics

Large list of math blogs. Highly recommended in particular are Terence Tao's and Tim Gowers'.

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The link is broken and now we have mathblogging.org. –  Junyan Xu Dec 20 '11 at 2:28
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Zentralblatt-MATH

MathSciNet has been listed above, but I didn't see Zentralblatt-Math. It does much the same thing as MathSciNet, although it has in fact been doing it far longer. Most papers get reviewed on both databases, and this redundancy if often very useful (although people frquently argue about whether or not we really need both nowadays - there have been many long discussions about this in various places).

Unfortunately, many students these days seem not to be aware of Zentralblatt. It is definitely a useful resource and if your institution pays for a subscription then it is certainly worth knowing about it and using it.

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David Ben-Zvi takes electronic notes on the talks he attends and posts them publicly. This can often be the best source of information for a subject which has not yet been written down.

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http://maths.dept.shef.ac.uk/magic/index.php

Apparently UK has been building a depository/interactive system for graduate math courses. Click on "courses" to access archives. Many have lecture notes and other materials.

I found this recently. Have not actually personally used it, but potentially very useful.

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A similar "access grid" network comprises the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. The main site (which includes links to archived course material) can be found at tcc.maths.ox.ac.uk. –  Nicholas Jackson Oct 24 '09 at 11:04
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