I was wondering if there is a list of the most active branches of mathematics?

If MathOverflow is a representative sample, then algebraic geometry is by far the most popular. Is this the case?

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    $\begingroup$ This is not an appropriate question to MathOverflow. Please read the FAQ before posting so that you get an idea of what is a good question: mathoverflow.net/howtoask.html $\endgroup$ Jan 5 '10 at 12:09
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    $\begingroup$ I disagree. Phrased more sharply (e.g. along the lines of my preliminary response below) it is appropriate: it is a question which is of interest to mathematicians and which has at least one answer. May we keep it open a little while to see what transpires? $\endgroup$ Jan 5 '10 at 12:20
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    $\begingroup$ OK -- the rephrased question is more sensible. I objected to the "fashionable" mostly. And also from the assumption that MathOverflow is somehow a representative sample of mathematicians. $\endgroup$ Jan 5 '10 at 12:30
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    $\begingroup$ This question (before and after editing) in not appropriate for mathoverflow. This is also inaprpriate editing since the edited question is rather different (and less interesting). $\endgroup$
    – Gil Kalai
    Jan 5 '10 at 12:44
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    $\begingroup$ I don't see anything wrong with this question (as edited). It is somewhat vague, because we have to decide whether "most" means "most papers", "most pages", "most researchers" or what, but it is not so vague that there aren't good ways of addressing it. Andrew and Jose's answers gave me a better understanding of the landscape of mathematical research. $\endgroup$ Jan 5 '10 at 19:19

Sorry to add to the noise, but here it goes. With a little script-fu (and emacs, of course!) I retrieved the data from MSC corresponding to the last ten years in each of the Primary Classifications. Annoyingly the AMS changed their subject classification scheme recently, so that the numbers I queried were interpreted as MSC2010, whereas the papers are published from the year 2000.

43465     35 Partial differential equations
38151     62 Statistics
35994     81 Quantum theory
35633     68 Computer science
34474     65 Numerical analysis
28593     05 Combinatorics
28296     90 Operations research, mathematical programming
26406     34 Ordinary differential equations
26192     60 Probability theory and stochastic processes
23879     93 Systems theory; control
22361     11 Number theory
21689     76 Fluid mechanics
20787     91 Game theory, economics, social and behavioral sciences
19440     37 Dynamical systems and ergodic theory
18425     83 Relativity and gravitational theory
17323     94 Information and communication, circuits
17247     53 Differential geometry
16465     47 Operator theory
16134     03 Mathematical logic and foundations
15408     20 Group theory and generalizations
14225     92 Biology and other natural sciences
14051     82 Statistical mechanics, structure of matter
13663     46 Functional analysis
12894     74 Mechanics of deformable solids
11241     14 Algebraic geometry
10237     49 Calculus of variations and optimal control; optimization
10215     30 Functions of a complex variable
10154     16 Associative rings and algebras
 9801     01 History and biography
 9781     54 General Topology
 8014     42 Fourier analysis
 7103     58 Global analysis, analysis on manifolds
 6780     15 Linear and multilinear algebra; matrix theory
 6410     70 Mechanics of particles and systems
 6359     32 Several complex variables and analytic spaces
 6348     57 Manifolds and cell complexes
 6185     41 Approximations and expansions
 5935     39 Difference and functional equations
 5684     26 Real functions
 5349     17 Nonassociative rings and algebras
 5226     13 Commutative rings and algebras
 4840     78 Optics, electromagnetic theory
 4439     52 Convex and discrete geometry
 4418     33 Special functions
 4350     00 General
 3818     06 Order, lattices, ordered algebraic structures
 3511     28 Measure and integration
 3295     51 Geometry
 2948     22 Topological groups, Lie groups
 2944     55 Algebraic topology
 2538     86 Geophysics
 2089     45 Integral equations
 2052     18 Category theory; homological algebra
 1679     80 Classical thermodynamics, heat transfer
 1523     31 Potential theory
 1444     43 Abstract harmonic analysis
 1343     12 Field theory and polynomials
 1161     40 Sequences, series, summability
 1108     08 General algebraic systems
  898     44 Integral transforms, operational calculus
  775     19 K-theory
  534     85 Astronomy and astrophysics

Usual disclaimers apply. In particular, before concluding that nobody works in astrophysics, go and check the submission statistics for astro-ph: more than 11,000 submissions in 2009 alone! Clearly the AMS does not index very widely in this area.

Let me reiterate that I do not believe for a second that this data allows one to conclude anything of value about mathematics, just perhaps about mathematicians :)

Added (incorporating Gerald Edgar's summary in the comment below)

This is the summary of "pure maths" defined as classifications 00-60, with a total of 411902 articles reviewed in the decade that has just finished. That, in case you are wondering is 55.38% of all papers reviewed.

00--08      Logic and Combinatorics         63804    15.49%
11--20      Algebra and Number Theory       80689    19.59%
22--49,60   Analysis and Probability       216252    52.50%
51--58      Geometry and Topology           51157    12.42%
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    $\begingroup$ Wow. That's very impressive. I'm also amazed that PDE really does dominate. This is consistent with the comment I made earlier about how top PDE people get way more citations than top people in other fields. $\endgroup$
    – Deane Yang
    Jan 5 '10 at 15:26
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    $\begingroup$ a summary... for what it's worth... Classifications 00--60, PURE MATH 411902 Classifications 00--08, LOGIC AND COMBINATORICS 63804, 15.49 percent of pure math Classifications 11--20, ALGEBRA AND NUMBER THEORY 80689, 19.59 percent of pure math Classifications 22--49 and 60, ANALYSIS AND PROBABILITY 216252, 52.5 percent of pure math Classifications 51--58, GEOMETRY AND TOPOLOGY 51157, 12.42 percent of pure math $\endgroup$ Jan 5 '10 at 18:28
  • $\begingroup$ Cool. There is no doubt that this is interesting information...and, of course, it confirms that I was right in my response below. $\endgroup$ Jan 5 '10 at 19:28
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    $\begingroup$ Ten years have passed. Is there any chance that we update this information? $\endgroup$ May 26 '20 at 21:12

For those who have time to do some coding, the AMS releases tables of how many papers in MathSciNet land in each of the MSC subjects. This should be a more representative sampling of mathematical publications than the arXiv. Unfortunately, the format is a list of every paper, its year of publication, and which classifications it used, so it is not obvious to a human which subjects are the most popular.

For those who don't have the energy to create our own table, David Rusin has a chart where the area of each MSC subject is proportional to the number of publications in that filed from 1980-2000. The classification is too fine to easily answer questions like "Is analysis more popular than algebra" and the time period is not quite what we want. But one can immediately see that any one of Statistics (62), Probability and Stochastic Processes (60), Numerical Analysis (65) and PDEs (35) all dwarf Algebraic Geometry (14), Category Theory (18) and even Number Theory (11).

  • $\begingroup$ Thanks -- that's an interesting link. I knew that the MathSciNet statistics were available, but I had misplaced the link. $\endgroup$ Jan 5 '10 at 14:38
  • $\begingroup$ @DavidSpeyer The link to David Rusin's chart is no longer availvable. $\endgroup$ Nov 27 '16 at 22:14

"Fashionable" is so subjective that it should be avoided here, I think.

On the other hand, it is very natural to wonder about which subject areas -- as represented, say, in the 2010 AMS Mathematics Subject Classification -- are the most popular as measured e.g. in terms of total papers published in the last ten years or the total number of mathematicians who have published in this area.

I'm not about to try to implement a computer search to answer this question, but it seems likely that someone else has already done so. I will predict an answer though: algebraic geometry is not the most popular research area in any quantitative sense. (Others have asked why algebraic geometry is so prevalent on MO and the most convincing answer seems to be that the founders of MO are mostly algebraic geometers and mathematicians in closely related areas.) I would be willing to bet that, as has been the case for at least one hundred years, more papers are published in analysis than in any other area.

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    $\begingroup$ This sort of quantitative measure has a clear bias, to which you are already alluding in your last sentence: "as has been the case for at least one hundred years". Since areas of research are, in most cases, inherited and since in many countries hirings are still very much "endogamous" (meaning people tend to hire their students), you can see how this sort of "popularity" can be maintained. I honestly do not think that this numbers game can teach us much. $\endgroup$ Jan 5 '10 at 12:34
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    $\begingroup$ It would be interested to see the statistics on average numbers of citations per paper as well. And Pete is correct- someone is interested in these measures. Canadian funding agencies go through this exercise every two years to assess which field to fund, and how much. Crude, disgusting, but there you have it. $\endgroup$ May 27 '11 at 14:31

The word "current" is my get-out on this! Here's the data from the arXiv for the month of december:

118 math-ph
111 math.PR
111 math.DG
 97 math.AG
 96 math.NT
 91 math.CO
 87 math.AP
 71 math.DS
 45 math.GR
 43 math.RT
 35 math.FA
 32 math.GT
 31 math.OC
 30 math.ST
 30 math.QA
 30 math.CA
 28 math.AT
 26 math.CV
 25 math.AC
 24 math.RA
 23 math.SG
 22 math.NA
 19 math.OA
 17 math.LO
 16 math.MG
 12 math.GM
 11 math.HO
 11 math.CT
 10 math.KT
  8 math.GN
  6 math.SP

(yeah, yeah, I know - skewed results since it came from the arXiv ... yawn, think of a new complaint, please.)

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    $\begingroup$ New complaint? Nah. I'll go with the one you gave yourself. (Why not skew it some more and just count tags on MO?) $\endgroup$ Jan 5 '10 at 14:05

I think it is a little bit anachronistic to divide mathematical disciplines and search for the most "active" one. The modern tendency (justified by the major achievements of contemporary mathematics) is to ignore the "barriers" between the different fields and become truly interdisciplinary.

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    $\begingroup$ I agree. The undergraduate curriculum would get a lot more done if it stressed this from the get-go. $\endgroup$ Jan 5 '10 at 14:33
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    $\begingroup$ Dear Anonymous, I'd like to be as optimistic as you, but I have the sad feeling that the barriers between different fields are getting higher, not lower. Whereas Euler could make fantastic contributions to fluid mechanics and number theory (among many, many other domains), I'm afraid it is improbable that a specialist in the classification of finite groups could win the Clay prize for the Navier-Stokes equations. I hasten to say that I'm equiignorant in both domains and that I would love to be proved wrong... $\endgroup$ Jan 5 '10 at 22:05

I'm skeptical that this question can be asked and answered in a meaningful manner. Do we really want to know which area of mathematics produces, say, the most papers? Or even the most citations? What might be more meaningful (but maybe not) is which fields get the most funding from NSF.

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    $\begingroup$ By the way, my highly unscientific investigations have led me to believe that among top mathematicians those in PDE's get by far the highest number of citations. $\endgroup$
    – Deane Yang
    Jan 5 '10 at 12:33
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    $\begingroup$ "Do we really want to know which area of mathematics produces, say, the most papers?" Sure, why not? $\endgroup$ Jan 5 '10 at 12:35
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    $\begingroup$ Pete is absolutely right: if someone is not interested in that sort of statistics and finds them meaningless, that's fine by me. But why should others be prevented from knowing them if they believe these numbers say something interesting about our community ? $\endgroup$ Jan 5 '10 at 21:47

Given the hint from David, here's what 30s coding produces:

 3627 35
 2853 81
 2355 05
 2228 68
 2192 34
 2172 94
 2083 76
 1985 11
 1852 60
 1752 65
 1728 90
 1676 83
 1657 37
 1639 53
 1550 91
 1413 47
 1373 20
 1362 93
 1356 03
 1256 82
 1187 74
 1184 62
 1151 46
 1113 14
 1033 92
  915 16
  778 49
  762 30
  714 42
  678 58
  588 57
  585 54
  534 32
  531 17
  521 39
  516 70
  504 41
  481 26
  476 13
  421 15
  375 33
  366 52
  323 06
  266 51
  246 22
  235 78
  235 55
  225 86
  193 28
  181 01
  178 00
  175 18
  171 80
  137 45
  133 31
  119 43
   95 12
   87 08
   86 19
   83 40
   68 44
   47 85

As it's only 30s, I'll leave it to others to fill in the data about which area is which MSC. (Community wikied so that others can easily do that). Oh, it's the 2007 data (most recent) by the way.

(yeah, yeah, I know - skewed results since it came from the MSC ... yawn, think of a new complaint, please.)


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