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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3$\begingroup$ 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 $\endgroup$– José Figueroa-O'FarrillCommented Jan 5, 2010 at 12:09
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15$\begingroup$ 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? $\endgroup$– Pete L. ClarkCommented Jan 5, 2010 at 12:20
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1$\begingroup$ 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. $\endgroup$– José Figueroa-O'FarrillCommented Jan 5, 2010 at 12:30
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2$\begingroup$ 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 KalaiCommented Jan 5, 2010 at 12:44
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4$\begingroup$ 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. $\endgroup$– David E SpeyerCommented Jan 5, 2010 at 19:19
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7 Answers
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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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3$\begingroup$ 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. $\endgroup$ Commented Jan 5, 2010 at 15:26
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2$\begingroup$ 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 $\endgroup$ Commented Jan 5, 2010 at 18:28
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$\begingroup$ Cool. There is no doubt that this is interesting information...and, of course, it confirms that I was right in my response below. $\endgroup$ Commented Jan 5, 2010 at 19:28
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8$\begingroup$ Ten years have passed. Is there any chance that we update this information? $\endgroup$ Commented May 26, 2020 at 21:12
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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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$\begingroup$ Thanks -- that's an interesting link. I knew that the MathSciNet statistics were available, but I had misplaced the link. $\endgroup$ Commented Jan 5, 2010 at 14:38
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$\begingroup$ @DavidSpeyer The link to David Rusin's chart is no longer availvable. $\endgroup$ Commented Nov 27, 2016 at 22:14
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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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11$\begingroup$ New complaint? Nah. I'll go with the one you gave yourself. (Why not skew it some more and just count tags on MO?) $\endgroup$ Commented Jan 5, 2010 at 14:05
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"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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2$\begingroup$ 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. $\endgroup$ Commented Jan 5, 2010 at 12:34
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1$\begingroup$ 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. $\endgroup$ Commented May 27, 2011 at 14:31
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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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1$\begingroup$ I agree. The undergraduate curriculum would get a lot more done if it stressed this from the get-go. $\endgroup$ Commented Jan 5, 2010 at 14:33
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15$\begingroup$ 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... $\endgroup$ Commented Jan 5, 2010 at 22:05
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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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2$\begingroup$ 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. $\endgroup$ Commented Jan 5, 2010 at 12:33
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3$\begingroup$ "Do we really want to know which area of mathematics produces, say, the most papers?" Sure, why not? $\endgroup$ Commented Jan 5, 2010 at 12:35
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2$\begingroup$ 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 ? $\endgroup$ Commented Jan 5, 2010 at 21:47
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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.)