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I have been involved in a number of large-ish research consortia usually with 10 or more groups, with each group containing 2 or more individuals. Some have originated from computer science, others engineering, and even the social sciences. In general all have contained partners from a variety of disciplines.

I'm curious as to how large mathematics research projects tend to be? Do they tend to be smaller, with individuals rather than research groups and/or subject matter experts involved. Are there any mathematical research associates out there that could offer insight into this please?

EDIT: Just to clarify, if P is a project, the "size" of P I'm interested in is:

|P|=Total number of people involved.
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It depends on what you mean by "projects", but there are both large and small ones.

There are both casual collaborations between two or three individuals, which translate to a single paper, and which can be funded individually (there are grants as small as 1k USD or less, the money for a single visit or a conference participation), and larger ones that can let you hire multiple people for several years (e.g., the EU grants called ERC, which can give six-digit amounts).

I have also seen even larger "projects", but they were usually structured as mini-funding agencies, giving out their funding to smaller individual research endeavours.

  • So would it be possible in mathematics to have a successful career soley being involved in these small "projects"?, i.e. 1-3 individuals. – Pixel Mar 8 '15 at 9:01
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    @pbs Definitely yes. Mathematics has very little lab expenses, so there is much less need for a single, coherent source of funding. A mathematician's career can revolve around single-name papers and casual temporary collaborations with colleagues met at conferences. – Federico Poloni Mar 8 '15 at 9:06
  • But of course (except for exceptional people like Erdös), in addition to these research projects, you would have a teaching position to provide your salary. – GEdgar Mar 8 '15 at 17:47
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It's rare for a large number of mathematicians to collaborate on a single project. The vast majority of mathematics papers have 1, 2 or 3 authors.

Using the techniques described in Getting a dump of arXiv metadata, I downloaded metadata for all the math preprints on arXiv, and had a script count the number of authors. I got the following results:

251459 papers
0 authors:      1 papers         0.0004%
1 authors:      101926 papers    40.5338%
2 authors:      89630 papers     35.6440%
3 authors:      43290 papers     17.2155%
4 authors:      12555 papers     4.9929%
5 authors:      2949 papers      1.1728%
6 authors:      700 papers       0.2784%
7 authors:      216 papers       0.0859%
8 authors:      102 papers       0.0406%
9 authors:      40 papers        0.0159%
10 authors:     14 papers        0.0056%
11 authors:     9 papers         0.0036%
12 authors:     7 papers         0.0028%
13 authors:     4 papers         0.0016%
14 authors:     2 papers         0.0008%
15 authors:     3 papers         0.0012%
16 authors:     2 papers         0.0008%
20 authors:     1 papers         0.0004%
22 authors:     3 papers         0.0012%
23 authors:     1 papers         0.0004%
28 authors:     1 papers         0.0004%
37 authors:     1 papers         0.0004%
60 authors:     1 papers         0.0004%
67 authors:     1 papers         0.0004%

I took a look at some of the outliers. Some are ordinary papers but some are other kinds of collective works. Note that some appear to be cross-listed under other arXiv sections.

To the asker: There may be some ambiguity here based on what we think of as a "project" and "people involved", and how research is conducted in our fields. As a mathematician, I think of a "project" as a concerted effort by a researcher, or group of researchers, actively working to resolve a specific mathematical question. If this effort is successful, it normally results in one or more published papers (sometimes three or four, but usually not dozens), and all the researchers who significantly contributed to the solution will be authors of those papers. If it's a two-author paper, then there really were only two people working on that project. (Of course, there could have been other unrelated groups working on the same question, or something closely related, but they would normally publish separately.) So to me, there is a pretty close correspondence between "project" and "paper".

You might also find this question interesting: How do mathematicians conduct research?

  • Thank you, that's interesting data. From my current perspective I'm not sure if number of authors is the best indicator though. For example, although my group (of size 3) is involved in a larger consortium we would only publish our own work under our names - unless of course someone else had helped with the work. It will be interesting to see the results of your larger datasets though ! This might be getting to the essence of my question: are maths research "projects" usually small (1-2 individuals involved), and as you say it is rare for more than 3 to collaborate. – Pixel Mar 8 '15 at 9:02
  • +1, really curious to see more of these data when you update (and also very curious about the 22 author paper!) – Aru Ray Mar 8 '15 at 12:48
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    Nate, I have already downloaded it and feel free to use them is it could save some of your time: all math metadata turned into csv and all metadata xml dumps. – Piotr Migdal Mar 8 '15 at 15:43
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    @AruRay: Updated! – Nate Eldredge Mar 8 '15 at 16:03
  • @PiotrMigdal: Thanks very much for the offer. I had already finished downloading it myself, but at least it was interesting to learn how that process works. – Nate Eldredge Mar 8 '15 at 16:04
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How common it is for mathematicians to be officially organized in structured projects varies a lot, depending on the country, the institution, and the type of mathematics.

For pure mathematics in U.S. universities, there's practically no notion of named projects or consortia. Just about everyone is an individual researcher, including graduate students. You may have short or long-term collaborations, and it's common to write papers together, but there are no formal organizations behind the collaborations. Even if a funding agency conceives of a grant as funding a specific project with PIs in charge of a group, it's not likely to be described this way in casual communication among mathematicians. You might say "Smith and I have been working a lot on widget optimization recently", but you wouldn't describe it as the WIDGEMAX project.

  • Thanks for the answer. From my experience (in the UK) our group uses the names to distinguish between projects as usually there will be a number of projects either active or in the developmen stage at any one time (these are none mathematical projects). So far I like what I'm hearing about mathematical collaboration and project work. – Pixel Mar 8 '15 at 17:22
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Are you asking about projects or papers? (You could measure how big a project is in person-hours, people/project, actual time/project, etc.) What I think of a project may be something spanning many years (possibly most of one's career) with lots of different papers, possibly with different sets of coauthors, or it could be a single standalone paper, or something that never turns into a paper at all. I would guess that most published research papers are pieces of a larger project that the investigator(s) are working on. For example, this is probably true for 2/3rds or 3/4th of my papers, depending how you count. The reason is because math is hard, and one typically can't prove what one hopes right away.

If you just are asking about number of people on papers this is easier to answer. There are two papers of Jerrold Grossman just about math publications you may find interesting, though the information is a bit dated (particularly compared to Nate's answer, but using Mathematical Reviews data, which is more representative than the arXiv):

These study things like trends in the number of coauthors and variation by field. For example, from Table 3 in the first article the average number of authors/paper went from 1.1 in the 1940's to 1.63 in the 1990's, and the number of 3+ author papers went from 1% to 13%. See also Figure 6 of the second article which has a graph. Table 3 of the second article, breaks things down by section classifications (which are subjective, but still instructive). Here is a sample from that table:

Section     #authors/paper   2+ authors  3+authors
CS                  1.77       53%       17.7%
Combinatorics       1.64       46%       17.7%
Statistics          1.56       45%        8.7%
Geometry            1.34       28%        4.9%
Number Theory       1.32       26%        5.0%

Note these numbers are for papers in between 1980 and 1999, so I suspect the numbers should be considerably higher now, based on the upward trends in Grossman's papers and just from what I personally see in Number Theory. This is also supported by the data from Nate's answer.

(Edit) I just remembered this paper (with 8 authors) which has more recent data than Grossman's (going up to 2009). See Figs 2a and 3a, from which it's clear that there's more collaboration in applied areas. The most recent data in there gives about 2.2 authors/paper for applied math and 1.8 for pure math. (End edit.)

For a comparative study across different disciplines, his paper of Newman is perhaps the most well known (at least to me) study. From Table 1 we see

discipline    avg #authors/paper
biology       3.75
physics       2.53
math          1.45
  • Thanks for your answer. My chosen measure of project size is |P|=Total number of people involved. – Pixel Mar 8 '15 at 14:14
  • Aha, this sounds much more like what I would expect. – Nate Eldredge Mar 8 '15 at 14:54
  • @NateEldredge Yes, thanks for catching the discrepancy. (I wasn't thinking when I typed my original answer---those numbers didn't make sense at all!) – Kimball Mar 8 '15 at 16:02

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