The "MathMap" Image

[Improved picture of the subfields of mathematics] [Legend: grouping of the subfields of mathematics]

The creation and interpretation of the MathMap

This image shows a disk for each of the 61 two-digit subject headings of the 1991 Mathematics Subject Classification. Each has area approximately proportional to the number of entries in the Math Reviews database 1980-1997 which have that area listed as primary classification. Their placement in the plane is intended to be a suggestion of the extent of their interaction with each other.

An "interactive" version of the MathMap allows (with an appropriate browser) the titles of the disciplines to "pop up" with a pointer motion, and provides links to an introduction to each area.

The determination of those positions is part of a project which I am currently undertaking. For the most part they were determined by objective criteria -- using the correlation between the primary and secondary classifications of items in the MR database, one can determine principal coordinates in the 61-dimensional "MSC-space", the principal useful 2-dimensional projection of which is the image shown. I have modified the positions of a handful of the areas in two ways: outliers have been moved inward to reduce wasted space, and overlapping disks have been spread apart into the nearest available cavity to avoid obscuring each other. Of course the decisions of how to do this are subjective. (The most noticeable change was that of the Probability (60) and (especially) Statistics (62) icons, which in MSC space lie essentially directly over the origin.)

Of course, the initial classification of MR items into the 61 areas is subjective too, but reflects the considered opinion of the many reviewers. In the coloring of the 61 disks and in the legend at the right, I have grouped the areas further into major headings; this reflects personal preference, certainly, but much of it (particularly the grouping of the fields of analysis) is suggested by the MathMap itself.

(Since the creation of the Math Map, the 2000 MSC has been issued. There are some changes in top-level classifications: 04 is gone, 73 is replaced by 74, 37 and 91 have been created from associated portions of other classifications, and a new 97 has been added for use as a secondary classification.)

Using the MathMap as a tool to correlate parts of mathematics

I have a number of projects in mind which would improve this picture; when completed, they will, I hope, make a nice article for publication. Keith Dennis, editor of Math Reviews, has been very cooperative and provided the necessary summary data; we are working to arrange more detailed data which should permit the creation and analysis of a number of related illustrations.

Any set of papers in MR can be demonstrated graphically on the MathMap; for example, you may wish to try identifying the eight most recent Fields Medalists from their set of papers. (It is interesting to note that Fields Medals appear not to be awarded for work distributed uniformly around the mathematics spectrum! Here is an illustration of the Fields' Medalists centers of mass.)

These images are taken from a talk I recently gave on this map; inquire for details.

Some 3D texturing of the image was performed with Mathematica. I have finally understood the graphics features of Mathematica well enough so that I can get the kind of display I ask for; now I need to think more carefully about what to ask for! Suggestions regarding lighting, colors, 3D effect, and so on are welcome.

The presentation and appearance of the MathMap

The representation of an image is handled differently with various combinations of hardware and software. This image is a 256-color GIF file. It may or may not appear appropriately on your devices, for several reasons:

I believe this file format is comprehensible to most web browsers (except text-based ones such as Lynx, of course) and can be processed with many image-viewing programs as well. I can translate it to some other formats for you if, for example, you have no way to view pictures except via the Windows Paintbrush program.

The presentation of various colors varies considerably on various machines. Certainly if you have access to 16 or fewer colors, this image is likely to look strange. Even some PCs here which can handle millions of colors (24-bit) process these colors very poorly; for example, the legend image comes out in grayscale, and the disks for areas 60 and 62 are of indistinguishable colors. I'm looking into this; I would appreciate hearing of your experience with the color separation. If you go to print this image, you may find that the colors are noticeably distinct from their screen appearance. (It is interesting to note that the hues used do not vary linearly, that is, it requires a very uneven separation of hue specifications in Mathematica to yield what appears on my devices, and to my eye, to be an even separation of apparent colors.)

Different sizes of the image are better for different applications. On the welcome page listed below, I have placed a small version of the mathmap. That image is as large as I can make it and still keep it within the width of the Netscape browser of all the machines at my disposal (it's 598 pixels wide; only 451 pixels high). If your browser fails to show the entire width of that image, I'd like to know so that I can put a smaller image on that web page. On the other hand, the smaller images become harder to read, of course; moreover, as I try to shrink font size less dramatically than image size (to improve legibility), the numerical labels begin to obscure much of the disks. So I'm providing larger images as well. The image at the top of this page is the largest I can make with my current combination of software packages (it's 726x809 pixels, nearly twice as tall as the other). My PC at home has a 1024x768 pixel display, so I've made an intermediate size version which just fills this screen (download and view with an image-viewer such as C-Show). At 100dpi resolution this nearly fills a US standard sheet of paper with less than half-inch margins, so you can have a copy of this image "suitable for framing", as they say. (Naturally the GIF image can be compressed or expanded, but this wreaks havoc with legibility and crispness, so adjustments in size have to be made manually with various Mathematica runs. If you would like an image of this map made to some custom size, let me know.)

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Last modified 2001/07/24 by Dave Rusin. Mail: