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The Mathematical Atlas is a collection of articles about aspects of mathematics at and above the university level, but (usually) not at the level of current research. The goal of this collection is to introduce the subject areas of modern mathematics, to describe a few of the milestone results and topics, and to give pointers to some of the key resources where further information is to be found. Like any good atlas, we try to present several ways to look at each area and to show its relationship with neighboring areas and sub-areas.

Sections of this document:

[Using what's here] [Forms of access] [External links] [New projects] [The MathMap]
[Site Statistics] [Reviews of this site] [Site History] [No Ads?] [Contacting us]

How to navigate through the Atlas

The Mathematical Atlas web site has been designed with the visitor in mind who wants to know something about a topic in mathematics -- why it's interesting, how it fits in with the rest of mathematics, and how it may be useful for solving some problem in or out of mathematics. Typically you, the visitor, will begin at the Welcome Page --- that's the page with the old map in the background. From there, you'll probably pass through a sequence of color-coded pages to zero in on what you're looking for, possibly linking to other sites along the way.

Note: Page colors are dependent on the color resolution of your display. The colors are intended to be unobtrusive and are not critical for successful navigation around the Atlas. Note that the color we call really means your browser's default.
Here is how your browser renders the other colors: Purple Green Blue White

Purple Navigation Pages

From the welcome page, you can choose any one of several methods to reach the material most of interest to you. There are several such navigation tools -- the purple pages -- from which to choose: according to taste, you can find your way around with

In each case, the goal is to lead you to the blue index page for the mathematical discipline most appropriate to your query.

If you select to search by keyword, you'll get a page offering several mathematics search engines. Several of them search databases here at the Mathematical Atlas, but several others search databases off-site. These are generally reputable sites, although we have not inspected all their material. If you are looking for a particular topic in mathematics but you're not sure how broad it is, begin with the search engines at the top of the list, which scan smaller databases. If you don't find sufficient hits, you may wish to try the larger databases, lower on the list.

If you select to browse a set of subject headings, you will be presented with a page listing about 100 topics from which to choose. One column contains the subject headings we use at this site in the blue index pages. Another column offers a few other schemes for indexing the parts of mathematics, although at this time only rudimentary cross-linking has been done to match the headings in other systems with the most appropriate index pages at this site.

If you select to use the MathMap to search for mathematics fields, you will find most of the same 100 topics. Here they are arranged with related areas close to each other, in case you know something about the branches of mathematics but you can't find a subject under the name you were expecting. Unless you are familiar with the numbering scheme of the MSC you will have to enable Javascript in your browser so that the pop-up boxes will show the subjects by name.

If you select to begin with the "tour" of the mathematical landscape, you will be led through a sequence of short descriptions of the major branches of mathematics, with increasing specificity, to lead you to the index page for the subject area which best matches your interest.

Every index page offers links to each of these families of navigation tools, as well as a "Help!" link which points to this page.

Blue Index Pages

The index pages are arranged thematically using the classification system of the subfields of mathematics used by the American Mathematical Society, Zentralblatt für Mathematik, and most other mathematical professionals. There is one index page for each of the (sixty-three) major headings of the Mathematics Subject Classification. There are also index pages for many of the (over 500) secondary headings, and a few of the (nearly 5000) tertiary headings of the MSC codes. Pages for more of the subject headings will be added over time.

Each index page has information to help you see what this portion of mathematics is about (and what it is not!). There is room for

The introduction and historical sketch should help clarify what some of the key ideas are; the next two are there to suggest good places to look if you're searching for something and didn't see it in the first two sections. Each of the field's sub-areas within the MSC code is listed, and there are many crosslinks among the index pages so that if the subject you're interested in is more carefully treated on the index page for an allied field, you'll be able to jump there.

Each page has a small "MathMap" showing the universe of papers 1980-1999 carrying a classification in this area. For example, the page for 30: Complex Analysis shows a map representing all papers bearing a classification '30'; their primary classification may be '30H' (Riemann surfaces), '30*' (textbooks, tables, etc.), '32' (Several Complex Variables), and so on. (Primary classifications accounting for less than 1% of the total are excluded.) As with the main MathMap on the Welcome Page, the sizes of the disks are unambiguously determined; their placement is not, but the current algorithms produce maps which agree moderately well with common opinion. In particular, the three or four clusters furthest from the main mass do usually represent cohesive and distinctive groupings of fields.

Note: the versions of the MSC used during the period 1980-1999 differ somewhat with the MSC2000 used for the organization of the index pages, so the set of subfields listed on the page need not agree with those shown in the maps. Explanatory information is included on the pages most affected.
The second half of the index page contains sources for further information, including These resources are in general expected to be nearly contiguous with the field of mathematics discussed just in this one index page. Materials of interest only for one or two subfields will be mentioned only on the index pages for those subfields; exceptions are sometimes made only as long as the index pages for the subfields have not yet been created. Remember too that any search for resources is only complete when the index page for the parent field is consulted as well.

We have tried to review the off-site web pages to which we provide links, but cannot monitor all changes in those sites; please notify us if links appear to have changed location, become unavailable, or seem inappropriate for this index page. Additional links can also be included but the editors reserve the right to limit the links to a small number of most useful resources. (The ideal situation occurs when another site provides a sorted comprehensive list of web links and other resources.)

You might get a better idea of what the index pages are to look like by glancing at the template for them. Room for future expansion of the set of index pages are provided; that is, most pages have dead links which can eventually point to further index pages on narrower topics, but which currently point to nothing of interest.

Green "Special Subjects" Pages

The MSC represents a common hierarchy for partitioning the topics in mathematics, but it is hardly the only choice. There are collections of MSC topics which are commonly treated as a cohesive unit; there are topics outside the scope of the MSC which are nonetheless arguably part of Mathematics; and there are topics in Mathematics which are too elementary or well-established to warrant a heading in the MSC. (Its primary purpose is for the classification of new research.) On occasion, we will provide pages similar to the blue index pages which discuss these topics.

At the present time there are few of these pages; you may view the complete list.

Gray "Selected Topics" Pages

The "Selected Topics", or "gray pages", offer further detailed information about specific topics within the given subject. No attempt has been made to balance these topics for uniformity with respect to subject coverage, level of sophistication, or style. The files are mostly flat text, taken from email, newsgroup posts, search engine outputs, bibliographies, external web pages, or special summary reports. You should set your browser to display these pages in a fixed-width font so that some of the multi-line formulas display properly.

A few of the the gray pages, and many documents at the sites to which we point from the index pages, are of some other filetype; neither flat text nor HTML is particularly well suited to transmitting mathematics. You may wish to equip your browser as is suggested for the readers of the newsgroup sci.math.research. For (free or cheap) TeX processors and DVI viewers, try the Tex Users' Group. (A quick-and-dirty alternative is this tex2mail Perl script.) For PDF and PS files you might want to follow these links for free software. You may find that some HTML files assume your browser to be capable of various newer features including subscripting and mixed fonts; you may need the latest version of Netscape, Internet Explorer, or Lynx.

The straight-text (ASCII) format is hardly ideal but meets the goal of being accessible to nearly every configuration of hardware and software. There are some informal conventions which allow us to work around the lack of appropriate symbolism which this approach entails; if you get stuck, you may wish to read this guide to informal math communication. (The index files, on the other hand, do use a few features which may not be available on all browsers; please report any difficulties you experience so they may be corrected.) For a short description of some of the formatting problems preventing the exchange of mathematics across the Web, you might wish to see the writing tips we follow. There is another document with examples and further discussion of various strategies for placing mathematics on the Web.

White PDF Pages

In response to the demands for accurate layout of mathematical notation we will be offering a number of the pages in Portable Document Format (PDF). You will need a PDF viewer for these files; they are freely available for most common computer platforms. These pages are intended to be identical in comment to the other pages listed above.

Some users have reported that some of the colored pages are displayed with white backgrounds, too. You may observe this because your hardware is unable to display the light tints used, or because your software has become unable to register the colors (in which case it is sometimes effective to close that application and re-launch your browser). In no case is the color representation essential for reading the documents; this is simply intended to help orient the visitor to our site.

How to access these files seamlessly

The materials in the Mathematical Atlas are freely available with nearly uninterrupted access. In order to take advantage of these materials, you need only arrange for a certain minimal configuration of resources.

System requirements: By design, the information content at this site is offered using only the most primitive features of web browsers. This should make the site (a) universally accessible, and (b) fast. In particular, we do not use animated GIFs, sound, movies, and so on.

Increasingly, however, we take advantage of some features available to web-page designers to improve presentation of these pages. We have tested these pages with various platforms (Wintel, Macs, Sparcs), browsers (Netscape, IE, Lynx), and display devices. Unfortunately, it is impossible to monitor every combination of hardware, operating system, internet connection, browser software (and user-selected options), and the fonts, colors, and display sizes of monitors and printers. In some cases your system configuration can distort the intended behaviour of the pages we provide.

Here are some of the features of these pages which might cause difficulty. If you have additional problems not discussed here, please send a description of the problem and of your system so that we can reformat the pages to alleviate the difficulty.

Special note regarding frames: We are currently experiencing some difficulties with the interpretations of frames by various browsers. If you enter the site from, say,

you may find that your browser fails to show accurate page titles and URLs in various display areas when you click to other pages, including pages off-site. If this causes any difficulties, you can reach the same material starting at the mirror site
(or more generally by replacing the domain-name www.math-atlas.org with the longer domain/directory string www.math.niu.edu/~rusin/known-math in any of our URLs). You will be presented exactly the same pages but in a frames-free manner. However, we request that you list the shorter math-atlas.org domain name in any public bookmarks or search engines.

Modes of access: The materials at this site can also be recovered using FTP. The file structure is just the welcome page and the files in these directories (there are no subdirectories):

(There are also some assorted bookkeeping and hashing files with restricted access.)

Since we expect substantial improvements to the Mathematical Atlas during 2000, we are not yet distributing the materials in bulk. (There is the possibility the pages can be collected into a CD-ROM in the future.) If you would like to set up a mirror for the files or need to secure compressed archives because of poor internet connections, please contact the site administrators.

Improvements in the organization and scope of this collection may occasionally require moving files around, so links to files in this collection should probably be limited to (or at least should include as back-up) the URL of the welcome page. In particular, users who made bookmarks to "gray pages" long ago may find the URLs no longer to be valid. Please send mail if you experience any difficulties.

Cross-links with other sites

Looking for something different from this site? Here are some recommended general-purpose external links:

For information at a more basic level than is covered in these files, read the Frequently Asked Questions in Mathematics file for the USENET newsgroup sci.math. You may prefer something less in-depth, such as an encyclopedia entry; for example you may begin with Encarta's Mathematics entry, or that of the Columbia Encyclopedia or Funk and Wagnall's Encyclopedia. If you feel more comfortable with material at the secondary school level or below, you might prefer something like the Math Forum. If instead you are interested in research-level aspects of mathematics, you are likely to be better served by the American Mathematical Society or the European Mathematical Society.

In general this site offers no unexamined laundry lists of websites and books. Should you seek large collections of links which intend to be comprehensive catalogues of the world-wide web, you may enjoy some of these points of access to mathematics web sites:

If instead you seek filtered directories which highlight the most appropriate resources, you might find one of these to be reasonably balanced in their coverage: Of course, an Internet-wide search engine can also provide links to mathematical material not mentioned at this site; while general-internet search engines can be useful (we have found both Google and Northern Light to be most intelligently organized), the search tools available at this site permit a more focused search for mathematical materials.

On the other hand, we offer quite a few links to appropriate subject-specific sites around the Internet: topic summaries, software, mailing lists, and so on. We cannot guarantee that these sites will be accessible when you try them, but the sites are verified periodically with webxref and other tools. We also cannot guarantee the quality of the material made available at the other sites, but an effort has been made to review these other sites; no links are made to sites found to have little scientific merit.

Conversely, this site is linked on a number of other pages (see e.g. the awards below). If you choose to link to this site, you may wish to use the 89 x 101 or 44 x 50 "MathMap" icon. Here is a suggested link:
[MathMap icon]
  <a href="http://www.math-atlas.org/welcome.html" target="_top">
  The Mathematical Atlas</a>:
  a gateway to the fields of modern mathematics

New projects

The index pages are incomplete right now (January 2000). Each of the 63 main areas of the MSC has an index page; pages for some of the sub-areas have also been created on an as-needed basis. There are 535 secondary areas and thousands of tertiary ones, so completeness is out of the question at this time! In many cases the index page is a nearly-empty shell right now, but this should be repaired during this year for the main areas. Pages for the subareas (e.g. 05C, Graph theory) will probably be improved after all 63 main pages. While the pages are in progress, it seems best to include more links to external sites than eventually will be the case, since it will take time to review them carefully.

We are completing a review of the mathematics literature 1980-1999 and will probably redraw the MathMaps to reflect this review. It is possible that we will render the MathMaps "live" to allow clicking to neighboring fields.

Other projects planned for 2000 include the addition of PDF options for some pages and the final conversion to the Year-2000 revision of the Mathematics Subject Classification. Internal changes will include a migration to TeX source code for HTML/PDF production, and a reorganization of the editorial board.

In addition to mathematical correctness and utility this site is monitored for appropriate performance under a variety of web-surfing conditions. Further tweaking is likely during 2000; please submit bug reports to the addresses below.

We hope to further advertise the site. The Mathematical Atlas is intended as a navigation tool for the visitor rather than a specialist's forum (hence the name). In various older web pages and search engines, this site is variously recorded as "Mathematics Essays" (reflecting the older emphasis on the Selected Topics files), "A Gateway to Mathematics" (still a subtitle!), and simply "Known Mathematics". Please share your interest in this site as math-atlas.org (note the hyphen). Advance apologies to international readers who might have preferred "maths-atlas".

The MathMap

The graphic image on the welcome page shows the top-level subareas of the Mathematics Subject Classification as a set of "bubbles". The surface area of each subfield of mathematics is proportional to to the number of recent papers in that area. The placement on the bubbles in the map is determined algorithmically by the frequency of cross-listing of papers in two or more areas. Thus it should be possible to locate related areas by selecting nearby bubbles on the map. This has already been done to group the subfields into sets for the tour.

We have made such maps available for subfields of the MSC as well; they are shown on the appropriate index pages.

These images are intended to offer a guide to a subject's placement in the mathematical universe. It is hoped that, for example, a visitor who cannot decide whether a question is part of Number Theory or Algebraic Geometry will visit one or the other of the index pages, and then note that, for example, section 11G of Number Theory is very close to Algebraic Geometry on the map for Number Theory; indeed a visit to the index page for 11G will then show that this area is an appropriate classification for work intermediate to those two areas.

These maps are intended to offer another way to navigate the Atlas, suitable for those who prefer to work graphically. If you have suggestions for alternative graphical interfaces for the Atlas, please write with your suggestions!

There's a separate page describing the creation and analysis of this "MathMap" image in greater detail.

Site Statistics

Here's a statistical abstract of the collection as of January 2000.

Currently there are about 113 (blue) index pages, as well as about 43 navigation (purple and green) pages, leading to about 2600 Selected Topics. (Many of the latter are compound files, made from about 5100 separate messages). Total size of the collection: about 21 Mb. This includes about a megabyte of indexing and organizational data, and about two megabytes of public information -- such as this file! -- describing aspects of the Atlas. The index pages include live links to about 900 other sites, including links on each page to appropriate pages at the AMS or EMS. There are about 1800 links between index pages, and over 4000 links between index pages and other resources at this site.

How well does the collection mirror the scope of mathematics? That depends in part on how one understands the layout of the mathematical landscape! That's subjective, of course, but for comparison, you might enjoy a comparison of sizes of subfields within mathematics (by some arbitrary standards). (Note that there is some imbalance among the areas in the MSC scheme.)

This site receives a visitor every 10 minutes or so to the welcome page (which should in general be an unduplicated count since only four files at this site offer a direct link back to it!)

It is difficult to measure the rate of "success" of visitors to this site. At the very least we hope that most of the visits do not lead to browser incompatibility, or other hardware/software issues which make the site less useful. We appreciate the supportive letters we have received from visitors, thank you!

Site History

The present Mathematical Atlas evolved from a much more informal collection of mathematical pointers collected by Prof. Dave Rusin of Northern Illinois University.

This site was originally designed to house a collection of short articles taken from USENET posts, mail, and various web sites. The collection was by done by me in a haphazard way. It was a combination of most of the substantive posts and email I had written, together with other items I considered useful (for me!), interesting, or well-written; it's best thought of as a public gateway to the files I thought worth saving for my personal use. So the material in these directories is heavily but not mostly my own work; this is intended to be neither plagiarism on the one hand nor self-aggrandizement on the other -- just useful. I believe the information in the files to be correct and germane, but I cannot always vouch for the accuracy of essays authored by others. Such a collection is also quite uneven in its coverage of the mathematics spectrum, of course. My own tastes tend towards algebra and geometry rather than analysis and applications. -- DJR

Here are a few links to files which describe the development of the collection of the "Selected Topics" files.

The initial collection of materials began in 1990 and was organized as a gopher site in 1995. With a simple index system this was launched as a web site March 1996. During 1997 the structure of the index pages was added but without much explanatory information. With the development of the MathMap in January 1998 it seemed appropriate to provide more thorough navigating information on the blue pages and to add several other navigating tools to serve visitors with different learning styles. During 1999 the local maps were added, an independent domain name was obtained, and the index pages were made more consistent regarding ISO encoding and cross-referencing; a substantial number of special topics files were included to broaden the areas of coverage somewhat. So far in 2000 we have rewritten the index pages to conform with MSC2000.

Reviews of this site

This site is included among the "basic mathematics resources" on many web pages. We particularly appreciate the recognition by comparatively selective sites and web directories such as those which bestowed the awards below:
[Braid of Links award] [Netscape]
[Key Resource award]
[Argus Clearinghouse Approved Site]
[NewHoo Cool Site]
[Education Planet Site]
CAMEL "Braid of Links"
Site of the Week
Netscape Directory Editors' Choice Links2Go Key Resource
Number Theory Topic
Links2Go Key Resource
Topology Topic
Argus Approved site
NewHoo CoolSite
Science/Math Resource
Education Planet
Top Site
Academic Resource Center
Awarded Site

According to http://www.websmostlinked.com/, this site ranks #74876 out of 446219 domains in their database of most linked sites. (yawn) You can rate this site:

Rate this site at Zeal.com

Even better: here's what they've said about this site around the world:

And, finally, an excerpt of a favorite post and a favorite review.

What are you selling here?

[NIU logo] The Mathematical Atlas gratefully acknowledges the Department of Mathematical Sciences at Northern Illinois University for the use of equipment and support personnel.

Apart from this acknowledgement, there are no ads at this site for ethical and aesthetic reasons.

The intent of the Atlas is to provide a balanced survey of the discipline of mathematics. It is difficult to claim an impartial perspective and a focused interest on the material while simultaneously displaying banners which are by nature intended to be self-serving and partial to one resource.

It must also be recognized that most advertising on the Internet -- perhaps by necessity -- is visually distracting and often unattractive. While the presentation of the material at this site is not as important as the subject matter it contains, it is well-known that educational materials which fail to provide an attractive "look" will be less effective at transmitting their content. We take the position that a non-commercial atmosphere is more appropriate for communicating subtle ideas.

"It is difficult to produce a television documentary that is both incisive and probing when every twelve minutes one is interrupted by twelve dancing rabbits singing about toilet paper." -- Rod Serling

A reasonable criterion for evaluating resources, particularly on the Internet, is to ask, "Why is this information being provided for free?". In the case of for-profit businesses, the case is fairly clear: the sponsoring organization wishes to provide whatever information will make it more likely that you will do business with them; a reputable corporation will in general provide information which is correct and complete if it complements their products. For example, a purveyor of cooking supplies may provide recipes which are appealing and whose preparation may involve their products in only a minor way; yet there is no reason to expect that the recipes offered will be nutritionally sound.

Correspondingly, one should ask whether the material at this or any other mathematics site is scientifically correct, well-balanced in coverage, and free of self-serving bias. In the case of mathematics, the material available on the Web is provided by (at least) three types of sources.

Of course, many sites represent a collective effort by many individuals; on the assumption that members of a collective will screen each other's work, it is probably true that such sites are at least as reputable as their most reputable member.

Now, if someone really wants T-shirts emblazoned with the graphics at this site. . .

Contacting us

This site only serves its purpose if you, the Internet visitor, can discover what you seek here. Please write us at rusin@math.niu.edu if you have any ideas for improvement (especially if you're willing to help turn them into reality . . .)

You can reach this page through the Welcome Page: http://www.math-atlas.org/welcome.html

Last modified 2002/03/14 by Dave Rusin, rusin@math.niu.edu You are also welcome to visit my home page, which includes links to other essays which are not strictly speaking mathematics -- mathematics teaching, applications of mathematics, mathematics contests, mathematical games, etc. -- as well as links to me and to my department.