# Quantitatively speaking, which subject area in mathematics is currently the most research active? [closed]

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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## closed as not constructive by José Figueroa-O'Farrill, Ilya Nikokoshev, Qiaochu Yuan, Reid Barton, S. Carnahan♦Jan 5 '10 at 16:15

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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 – José Figueroa-O'Farrill Jan 5 '10 at 12:09
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? – Pete L. Clark Jan 5 '10 at 12:20
Perhaps this should be community wiki, I tend to think that a question tagged "soft-question" should be community wiki. – Grétar Amazeen Jan 5 '10 at 12:29
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). – Gil Kalai Jan 5 '10 at 12:44
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. – David Speyer 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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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. – Deane Yang Jan 5 '10 at 15:26
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 – Gerald Edgar Jan 5 '10 at 18:28
Thanks. I've added to the main body of the answer. – José Figueroa-O'Farrill Jan 5 '10 at 19:12
Cool. There is no doubt that this is interesting information...and, of course, it confirms that I was right in my response below. – Pete L. Clark Jan 5 '10 at 19:28

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).

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Thanks -- that's an interesting link. I knew that the MathSciNet statistics were available, but I had misplaced the link. – José Figueroa-O'Farrill Jan 5 '10 at 14:38

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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New complaint? Nah. I'll go with the one you gave yourself. (Why not skew it some more and just count tags on MO?) – Harald Hanche-Olsen 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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I agree. The undergraduate curriculum would get a lot more done if it stressed this from the get-go. – Qiaochu Yuan Jan 5 '10 at 14:33
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... – Georges Elencwajg Jan 5 '10 at 22:05

"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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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. – José Figueroa-O'Farrill Jan 5 '10 at 12:34
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. – Nilima Nigam May 27 '11 at 14:31

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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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. – Deane Yang Jan 5 '10 at 12:33
"Do we really want to know which area of mathematics produces, say, the most papers?" Sure, why not? – Pete L. Clark Jan 5 '10 at 12:35
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 ? – Georges Elencwajg 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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