Bibliometric Analysis of Senior US Mathematics Faculty
BBIBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICSFACULTY
JOSHUA PAIK AND IGOR RIVIN
Abstract.
We introduce a methodology to analyze citation metrics across fieldsof Mathematics. We use this methodology to collect and analyze the MathSciNet( ) profiles of Full Professors of Mathematics atall 131 R1, research oriented US universities. The data recorded was citations, field,and time since first publication. We perform basic analysis and provide a rankingof US math departments, based on age corrected and field adjusted citations.
Introduction
For as long as the authors can remember, there has been discussion of compa-rable quality of various researchers (in all fields of research, but the authors aremost familiar with mathematics, so this paper concerns itself with mathematicsexclusively ). While such a comparison is not strictly speaking possible (mathemat-ics is not like competitive swimming, where a single number determines the betterswimmer), those of us who have been on hiring committees have needed to compareresearchers in diverse fields, and those of us who have had students (or job offers)have had to have some sort of estimate of the quality of people independently ofage and field (and building on that, to have some reasonably gauge of the qualityof departments ). The (admittedly ambitious) purpose of this note is to propose an objective metric, based entirely on citations data (as such, it can be gamed, as canany metric be).Briefly, we normalize the number of MathSciNet citations by dividing it by thenumber of years since the author’s first paper raised to the magical power 1 .
3. Wefurther segment mathematics into a number of “major fields”, assign mathematiciansto fields (this is very difficult for some people, including, ironically, the authors ofthis paper), and compute the z -score of the normalized citation number. For each de-partment we then compute the mean z -score of faculty to compute the department’sranking. Mathematics Subject Classification.
Key words and phrases. citations, ranking. a r X i v : . [ m a t h . HO ] A ug JOSHUA PAIK AND IGOR RIVIN Data
Data was collected between January 16 - January 24, 2020. After accessing the listof R1 schools, we found the faculty lists from the relevant departmental web page,determined their level of seniority, and searched their profiles on MathSciNet. In totalwe collected citation records for 2807 math professors at 131 different institutions.We then collected the total citations, the year of earliest indexed publication, and thefield of the most cited publication for each mathematician. A second pass throughthe data set occurred between January 30 - February 2, 2020, and discovered errorswere corrected. This data set is as complete as possible.There were initially 65 fields that were most cited as classified by MathSciNet,which we reduced to 20 fields, using the mapping in the appendix. This mappingwas constructed using general expertise on the way each field worked, and is recordedin the appendix. 2.
Exploratory Data Analysis
Distribution.
Citations and Citations/Year . appear exponentially distributedafter transformation by a square root.Figure 1: Distribution of citations and an exponential qq-plot of square rootcitations.Figure 2: Distribution of citations per year . and an exponential qq-plot of squareroot citations per year . . IBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICS FACULTY 3
Citations/Year . vs. Age. As noted in earlier work [2], citations and yearare positively correlated, but citations per year . and year are not correlated . Werepeat this analysis to check for robustness and find that when linearly regressed,citations/year . and year have a slope of 0 . − . , . p -value is 0 . R = 0 .
03. We conclude that citations/year . and year arenot correlated.Figure 3: Scatter plot of citations/year . and Years elapsed since first publication.The red line indicates the regressed line with equation0 . Y ear + 8 .
40 =
Citations/Y ear . . The p -value for slope is 0 .
122 and the R = 0 .
03. We fail to reject the null hypothesis that the slope is zero and concludethey are not correlated.2.3.
Factored Linear Models.
Before proceeding with the analysis, we shouldassess the importance of the explanatory variables when looking at differences incitations between mathematicians. We do this by constructing nested linear modelswith all three combinations of
Year and
Field , and determine the best model withAkaike Information Criterion, AIC [1]. Per standard interpretation the lower theAIC, the better the model. Let C =Citations, a = Age, and f = Field.Model AIC scorelog( C ) = β a + β f + (cid:15) . C ) = β f + (cid:15) . C ) = β a + (cid:15) . The exponent 1 . via a statistical analysis JOSHUA PAIK AND IGOR RIVIN in the RMarkdown on Github. It is clear that certain fields contribute negatively tooverall citations and other fields contribute positively.While it is not appropriate to pick a model for the sole reason that it minimizesAIC, it makes sense to consider both age and field.2.4.
Fields.
Different fields in mathematics have different citation practices. Somefields like Partial Differential Equations have more mathematicians, whereas somefields like Number Theory have fewer mathematicians. Some results from fields liketopology are widely applicable across disciplines, whereas more obscure results arenot. We quantify the bibliographic differences between fields. Note that the majorfields below are larger categories containing potentially multiple MathSciNet tags,and the mappings are recorded in the appendix.Major Field Mean Citations S.D. Citations Mean Cit/Year . CountPDE 1472.07 2182.45 14.58 372Computer Science 1260.44 2223.06 14.08 225Probability 1165.92 1401.07 12.06 137Harmonic Analysis 1120.12 1336.62 10.51 200Combinatorics 1023.24 1673.53 10.08 116Algebra 934.42 1310.59 9.12 220Algebraic Geometry 846.62 1308.80 9.51 169Geometry 890.68 1486.72 8.87 311Number Theory 742.66 920.31 7.38 159Dynamics 560.44 555.17 7.33 68Mathematical Physics 643.01 716.41 7.25 96Analysis 977.18 1951.95 7.15 45Applied Mathematics 646.60 976.98 6.87 299Group Theory 686.38 1151.64 6.74 81Logic 634.00 690.14 6.32 55Complex Analysis 612.86 725.15 6.17 115Lie Groups 512.02 590.59 4.78 43Statistics 220.73 331.15 3.10 83History 74.0 104.65 0.677 2Other 5 6.61 0.074 11Figure 5: Mean citations and citations per year . including counts, split by field,from top to bottom ranked by mean citations per year . to account for age.We ran a permutation test between each field to verify the observed partial order.We report the inconclusive differences ( p -value greater than 0 .
05) between fields whencomparing citations per year . in the appendix. IBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICS FACULTY 5
Ranking of Departments.
The above figures shows that comparing mathe-maticians in two different fields is akin to comparing apples and oranges. The cleanestway to standardize this is to compute an interfield z-score of the citations per year . ,and hence associating a “rank” to each mathematician. Then we computed the meanof the interfield z -scores of each full professor at the respective institution. We reporthere the top 20 schools and record the remaining schools in the appendix (1) Princeton University(2) Harvard University(3) Stanford University(4) University of Chicago(5) Columbia University in the City of New York(6) Massachussetts Institute of Technology(7) University of California, Los Angeles(8) University of Miami(9) Yale University(10) Brown University(11) University of California, Berkeley(12) New York University(13) University of Oregon(14) California Institute of Technology(15) Duke University(16) Stony Brook University(17) Rutgers University-New Brunswick(18) University of Virginia(19) Texas A&M University(20) Northwestern University 3. Conclusion
The rankings based on our normalized z -score (call it the PR score) correspondreasonably well with the “folk” rankings of mathematicians. While we do not wantto flatter or insult individuals by giving their scores here, we do give a ranking ofdepartments, and we see that it, again, corresponds well with the “folk” rankings. Ifthey do not, we encourage the reader to look at the faculty pages of the departmentsin question. It seems, therefore, that there is, indeed, a fully quantitative way toproduce meaningful rankings which work at least in a statistical sense - they fail forpolymaths, and they also are less successful for mathematicians the bulk of whose Where by ”Schools” we mean ”Mathematics departments” - for Universities with separate Pureand Applied math departments, the ranking will be different if the departments were to be combined.
JOSHUA PAIK AND IGOR RIVIN work is not indexed by MathSciNet - in particular those who do interdisciplinarywork.
References [1] Christopher M Bishop.
Machine learning and pattern recognition . 2006.[2] Joshua Paik and Igor Rivin. Data analysis of the responses to professor abigail thompson’sstatement on mandatory diversity statements, 2020. Appendix
Code and Data.
Available at https://github.com/joshp112358/Differences .4.2.
Classifications.
IBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICS FACULTY 7
Major Field Sub FieldsAlgebra Algebraic Topology; AssociativeRings and Algebra; Category theory, Homological algebra ;Commutative rings and algebras; Field theory;General algebraic systems; K-theory; Linearand Multilinear Algebra, matrix theory; Associativerings and algebras; Order, lattices, ordered algebraicstructures.Algebraic Geometry Algebraic GeometryAnalysis Difference and functional equations;Integral equations; Integral transforms, operational calculus;Ordinary differential equations; Real functions;Special functions.Applied Mathematics Approximations and expansions; Biologyother natural sciences; Calculus of variations and optimalcontrol, optimization; Fluid mechanics,Game theory, economics, social and behavioral sciences;Geophysics, Mechanics of deformable sciences;Mechanics of solids, Operations research, mathematicalprogramming; Systems theory, control.Combinatorics CombinatoricsComplex Analysis Functions of a complex variable; Potential theory; Severalcomplex variables and analytic spacesComputer Science Computer Science; Numerical Analysis;Information and communication, circuits.Dynamics Dynamical Systems and Ergodic TheoryGeometry Convex and discrete geometry; Differential Geometry;General topology; Geometry;Manifolds and cell complexes;Group theory Group theory and generalizations.Harmonic analysis Abstract harmonic analysis; Fourier analysis;Functional analysis; Global analysis, analysis on manifolds;Measure and integration, Operator theory.History History and biography.Lie Groups Topological Groups, Lie Groups.Logic Logic and foundations; Mathematical logic and foundations;Set theory
JOSHUA PAIK AND IGOR RIVIN
Major Field Sub FieldsMathematical Physics Classical thermodynamics, heat transfer;Mechanics of particles and systems; Optics, electromagnetictheory; Quantum theory; Relativity and gravitational theory;Statistical mechanics, structure of matter.Number Theory Number TheoryOther OtherPDEs Partial Differential Equations; Global Analysis,Analysis on manifoldsProbability Probability theory and stochastic processesStatistics Statistics4.3.
Ranking of Departments. (1) Princeton University(2) Harvard University(3) Stanford University(4) University of Chicago(5) Columbia University in the City of New York(6) Massachussetts Institute of Technology(7) University of California, Los Angeles(8) University of Miami(9) Yale University(10) Brown University(11) University of California, Berkeley(12) New York University(13) University of Oregon(14) California Institute of Technology(15) Duke University(16) Stony Brook University(17) Rutgers University-New Brunswick(18) University of Virginia(19) Texas A&M University(20) Northwestern University(21) University of Michigan(22) Rice University(23) The University of Texas at Austin(24) Carnegie Mellon University(25) University of Illinois at Chicago(26) University of California, Irvine
IBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICS FACULTY 9 (27) University of Pittsburgh(28) Georgia Institute of Technology(29) University of Minnesota(30) Vanderbilt University(31) Indiana University Bloomington(32) SUNY at Albany(33) University of California, San Diego(34) University of North Texas(35) University of Washington(36) University of Connecticut(37) Arizona State University(38) Pennsylvania State University(39) University of Southern California(40) Purdue University(41) University of Illinois at Urbana-Champaign(42) Cornell University(43) University of Maryland - College Park(44) University of Utah(45) North Carolina State University(46) Johns Hopkins University(47) University of California, Riverside(48) University of California, Santa Cruz(49) Washington University in St. Louis(50) Wayne State University(51) University of Pennsylvania(52) Brandeis University(53) Colorado State University(54) University of Notre Dame(55) University of California, Santa Barbara(56) University of North Carolina at Chapel Hill(57) University of Houston(58) University of Iowa(59) The Ohio State University(60) University of South Florida(61) Michigan State University(62) University of California, Davis(63) Virginia Polytechnic Institute and State University(64) University of Missouri(65) University of Wisconsin - Madison(66) University of Massachusetts Amherst (67) University of South Carolina(68) Emory University(69) University of Central Florida(70) University of Kentucky(71) University of Florida(72) University of Delaware(73) Louisiana State University(74) Syracuse University(75) Georgia State University(76) University of Colorado Denver(77) Boston University(78) Tulane University of Louisiana(79) Clemson University(80) University of Kansas(81) University of Southern Mississippi(82) Boston College(83) Mississippi State University(84) University of Rochester(85) CUNY Graduate School and University Center(86) The University of Tennessee(87) George Washington University(88) Georgetown University(89) Florida State Universty(90) Iowa State University(91) University at Buffalo(92) Northeastern University(93) Tufts University(94) University of Nebraska-Lincoln(95) University of Georgia(96) University of New Hampshire(97) Virgina Commonwealth(98) University of Cincinatti(99) Dartmouth College(100) Rennselaer Polytechnic Institute(101) University of Nevada, Reno(102) West Virginia University(103) Auburn University(104) The University of Texas at Arlington(105) Texas Tech University(106) University of Arizona
IBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICS FACULTY 11 (107) Binghamton University(108) University of New Mexico(109) The University of Alabama(110) The University of Texas at Dallas(111) George Mason University(112) Florida Institute University(113) University of Oklahoma(114) University of Colorado Boulder(115) University of Hawaii at Manoa(116) Case Western Reserve University(117) University of Alabama at Birmingham(118) Oklahoma State University(119) Kansas State University(120) Temple University(121) Oregon State University(122) Drexel University(123) University of Louisville(124) University of Nevada, Las Vegas(125) University of Wisconsin - Milwaukee(126) Washington State University(127) New Jersey Institute of Technology(128) The University of Texas at El Paso(129) University of Mississippi(130) University of Arkansas(131) Montana State University4.4.
Inconclusive Permutation Tests between Fields.
We report the results ofa one sided permutation test, when comparing cit/year . which failed to be signifi-cant at the 0 .
05 level. We record the hypothesis on the left and the p -value to theright.PDE ≥ Computer Science — p -value: 0.397PDE ≥ Probability — p -value: 0.0768Computer Science ≥ Probability — p -value: 0.156Probability ≥ Harmonic Analysis — p -value: 0.113Probability ≥ Combinatorics — p -value: 0.1049Harmonic Analysis ≥ Combinatorics — p -value: 0.3824Harmonic Analysis ≥ Algebra — p -value: 0.0961Harmonic Analysis ≥ Algebraic Geometry — p -value: 0.1807Combinatorics ≥ Algebra — p -value: 0.2181Combinatorics ≥ Algebraic Geometry — p -value: 0.3265 Combinatorics ≥ Geometry — p -value: 0.1544Combinatorics ≥ Analysis — p -value: 0.0719Algebra ≥ Algebraic Geometry — p -value: 0.6461Algebra ≥ Geometry — p -value: 0.3989Algebra ≥ Dynamics — p -value: 0.0813Algebra ≥ Mathematical Physics — p -value: 0.0546Algebra ≥ Analysis — p -value: 0.1232Algebraic Geometry ≥ Geometry — p -value: 0.2533Algebraic Geometry ≥ Analysis — p -value: 0.0782Geometry ≥ Number Theory — p -value: 0.0534Geometry ≥ Dynamics — p -value: 0.1076Geometry ≥ Mathematical Physics — p -value: 0.0704Geometry ≥ Analysis — p -value: 0.1497Number Theory ≥ Dynamics — p -value: 0.4906Number Theory ≥ Mathematical Physics — p -value: 0.4514Number Theory ≥ Analysis — p -value: 0.4417Number Theory ≥ Applied Mathematics — p -value: 0.2717Number Theory ≥ Group Theory — p -value: 0.2637Number Theory ≥ Logic — p -value: 0.1557Number Theory ≥ Complex Analysis — p -value: 0.0717Dynamics ≥ Mathematical Physics — p -value: 0.4614Dynamics ≥ Analysis — p -value: 0.4609Dynamics ≥ Applied Mathematics — p -value: 0.3377Dynamics ≥ Group Theory — p -value: 0.2999Dynamics ≥ Logic — p -value: 0.1738Dynamics ≥ Complex Analysis — p -value: 0.1272Mathematical Physics ≥ Analysis — p -value: 0.4916Mathematical Physics ≥ Applied Mathematics — p -value: 0.3506Mathematical Physics ≥ Group Theory — p -value: 0.337Mathematical Physics ≥ Logic — p -value: 0.23Mathematical Physics ≥ Complex Analysis — p -value: 0.1423Mathematical Physics ≥ History — p -value: 0.0525Analysis ≥ Applied Mathematics — p -value: 0.4035Analysis ≥ Group Theory — p -value: 0.3972Analysis ≥ Logic — p -value: 0.3154Analysis ≥ Complex Analysis — p -value: 0.2375Analysis ≥ Lie Groups — p -value: 0.1041Analysis ≥ History — p -value: 0.0881Applied Mathematics ≥ Group Theory — p -value: 0.4655Applied Mathematics ≥ Logic — p -value: 0.3609 IBLIOMETRIC ANALYSIS OF SENIOR US MATHEMATICS FACULTY 13
Applied Mathematics ≥ Complex Analysis — p -value: 0.2362Applied Mathematics ≥ Lie Groups — p -value: 0.0616Applied Mathematics ≥ History — p -value: 0.0534Group Theory ≥ Logic — p -value: 0.3728Group Theory ≥ Complex Analysis — p -value: 0.2834Group Theory ≥ Lie Groups — p -value: 0.0513Logic ≥ Complex Analysis — p -value: 0.4276Logic ≥ Lie Groups — p -value: 0.0613Complex Analysis ≥ Lie Groups — p -value: 0.0871Lie Groups ≥ History — p -value: 0.0531Statistics ≥ History — p -value: 0.2517History ≥ Other — p -value: 0.1533 School of Mathematics and Statistics, University of St Andrews
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