Bibliometric Analysis of Senior US Mathematics Faculty

Citations and Citations/Year^1.3 appear exponentially distributed after transformation by a square root.

Figure 1: Distribution of citations and an exponential qq-plot of square root citations.

Figure 2: Distribution of citations per year^1.3 and an exponential qq-plot of square root citations per year^1.3.

As noted in earlier work [2], citations and year are positively correlated, but citations per year^1.3 and year are not correlated¹¹1The exponent $1.3$ was determined via a statistical analysis. We repeat this analysis to check for robustness and find that when linearly regressed, citations/year^1.3 and year have a slope of $0.0338$ with a 95% confidence interval of $[-0.00899,0.0766]$. The $p$-value is $0.122$, so we fail to reject the null hypothesis that the slope is zero, and $R^{2}=0.03$. We conclude that citations/year^1.3 and year are not correlated.

Before proceeding with the analysis, we should assess the importance of the explanatory variables when looking at differences in citations between mathematicians. We do this by constructing nested linear models with all three combinations of Year and Field, and determine the best model with Akaike Information Criterion, AIC [1]. Per standard interpretation the lower the AIC, the better the model. Let $C=$Citations, $a=$ Age, and $f=$ Field.

While it is not appropriate to pick a model for the sole reason that it minimizes AIC, it makes sense to consider both age and field.

Different fields in mathematics have different citation practices. Some fields like Partial Differential Equations have more mathematicians, whereas some fields like Number Theory have fewer mathematicians. Some results from fields like topology are widely applicable across disciplines, whereas more obscure results are not. We quantify the bibliographic differences between fields. Note that the major fields below are larger categories containing potentially multiple MathSciNet tags, and the mappings are recorded in the appendix.

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 when comparing citations per year^1.3 in the appendix.

The above figures shows that comparing mathematicians in two different fields is akin to comparing apples and oranges. The cleanest way to standardize this is to compute an interfield z-score of the citations per year^1.3, and hence associating a “rank” to each mathematician. Then we computed the mean of the interfield $z$-scores of each full professor at the respective institution. We report here the top 20 schools and record the remaining schools in the appendix²²2Where by ”Schools” we mean ”Mathematics departments” - for Universities with separate Pure and Applied math departments, the ranking will be different if the departments were to be combined.

1. (1)

   Princeton University

2. (2)

   Harvard University

3. (3)

   Stanford University

4. (4)

   University of Chicago

5. (5)

   Columbia University in the City of New York

6. (6)

   Massachussetts Institute of Technology

7. (7)

   University of California, Los Angeles

8. (8)

   University of Miami

9. (9)

   Yale University

10. (10)

    Brown University

11. (11)

    University of California, Berkeley

12. (12)

    New York University

13. (13)

    University of Oregon

14. (14)

    California Institute of Technology

15. (15)

    Duke University

16. (16)

    Stony Brook University

17. (17)

    Rutgers University-New Brunswick

18. (18)

    University of Virginia

19. (19)

    Texas A&M University

20. (20)

    Northwestern University

Available at https://github.com/joshp112358/Differences .

1. (1)

   Princeton University

2. (2)

   Harvard University

3. (3)

   Stanford University

4. (4)

   University of Chicago

5. (5)

   Columbia University in the City of New York

6. (6)

   Massachussetts Institute of Technology

7. (7)

   University of California, Los Angeles

8. (8)

   University of Miami

9. (9)

   Yale University

10. (10)

    Brown University

11. (11)

    University of California, Berkeley

12. (12)

    New York University

13. (13)

    University of Oregon

14. (14)

    California Institute of Technology

15. (15)

    Duke University

16. (16)

    Stony Brook University

17. (17)

    Rutgers University-New Brunswick

18. (18)

    University of Virginia

19. (19)

    Texas A&M University

20. (20)

    Northwestern University

21. (21)

    University of Michigan

22. (22)

    Rice University

23. (23)

    The University of Texas at Austin

24. (24)

    Carnegie Mellon University

25. (25)

    University of Illinois at Chicago

26. (26)

    University of California, Irvine

27. (27)

    University of Pittsburgh

28. (28)

    Georgia Institute of Technology

29. (29)

    University of Minnesota

30. (30)

    Vanderbilt University

31. (31)

    Indiana University Bloomington

32. (32)

    SUNY at Albany

33. (33)

    University of California, San Diego

34. (34)

    University of North Texas

35. (35)

    University of Washington

36. (36)

    University of Connecticut

37. (37)

    Arizona State University

38. (38)

    Pennsylvania State University

39. (39)

    University of Southern California

40. (40)

    Purdue University

41. (41)

    University of Illinois at Urbana-Champaign

42. (42)

    Cornell University

43. (43)

    University of Maryland - College Park

44. (44)

    University of Utah

45. (45)

    North Carolina State University

46. (46)

    Johns Hopkins University

47. (47)

    University of California, Riverside

48. (48)

    University of California, Santa Cruz

49. (49)

    Washington University in St. Louis

50. (50)

    Wayne State University

51. (51)

    University of Pennsylvania

52. (52)

    Brandeis University

53. (53)

    Colorado State University

54. (54)

    University of Notre Dame

55. (55)

    University of California, Santa Barbara

56. (56)

    University of North Carolina at Chapel Hill

57. (57)

    University of Houston

58. (58)

    University of Iowa

59. (59)

    The Ohio State University

60. (60)

    University of South Florida

61. (61)

    Michigan State University

62. (62)

    University of California, Davis

63. (63)

    Virginia Polytechnic Institute and State University

64. (64)

    University of Missouri

65. (65)

    University of Wisconsin - Madison

66. (66)

    University of Massachusetts Amherst

67. (67)

    University of South Carolina

68. (68)

    Emory University

69. (69)

    University of Central Florida

70. (70)

    University of Kentucky

71. (71)

    University of Florida

72. (72)

    University of Delaware

73. (73)

    Louisiana State University

74. (74)

    Syracuse University

75. (75)

    Georgia State University

76. (76)

    University of Colorado Denver

77. (77)

    Boston University

78. (78)

    Tulane University of Louisiana

79. (79)

    Clemson University

80. (80)

    University of Kansas

81. (81)

    University of Southern Mississippi

82. (82)

    Boston College

83. (83)

    Mississippi State University

84. (84)

    University of Rochester

85. (85)

    CUNY Graduate School and University Center

86. (86)

    The University of Tennessee

87. (87)

    George Washington University

88. (88)

    Georgetown University

89. (89)

    Florida State Universty

90. (90)

    Iowa State University

91. (91)

    University at Buffalo

92. (92)

    Northeastern University

93. (93)

    Tufts University

94. (94)

    University of Nebraska-Lincoln

95. (95)

    University of Georgia

96. (96)

    University of New Hampshire

97. (97)

    Virgina Commonwealth

98. (98)

    University of Cincinatti

99. (99)

    Dartmouth College

100. (100)

     Rennselaer Polytechnic Institute

101. (101)

     University of Nevada, Reno

102. (102)

     West Virginia University

103. (103)

     Auburn University

104. (104)

     The University of Texas at Arlington

105. (105)

     Texas Tech University

106. (106)

     University of Arizona

107. (107)

     Binghamton University

108. (108)

     University of New Mexico

109. (109)

     The University of Alabama

110. (110)

     The University of Texas at Dallas

111. (111)

     George Mason University

112. (112)

     Florida Institute University

113. (113)

     University of Oklahoma

114. (114)

     University of Colorado Boulder

115. (115)

     University of Hawaii at Manoa

116. (116)

     Case Western Reserve University

117. (117)

     University of Alabama at Birmingham

118. (118)

     Oklahoma State University

119. (119)

     Kansas State University

120. (120)

     Temple University

121. (121)

     Oregon State University

122. (122)

     Drexel University

123. (123)

     University of Louisville

124. (124)

     University of Nevada, Las Vegas

125. (125)

     University of Wisconsin - Milwaukee

126. (126)

     Washington State University

127. (127)

     New Jersey Institute of Technology

128. (128)

     The University of Texas at El Paso

129. (129)

     University of Mississippi

130. (130)

     University of Arkansas

131. (131)

     Montana State University

We report the results of a one sided permutation test, when comparing cit/year^1.3 which failed to be significant at the $0.05$ level. We record the hypothesis on the left and the $p$-value to the right.

PDE $\geq$ Computer Science — $p$-value: 0.397

PDE $\geq$ Probability — $p$-value: 0.0768

Computer Science $\geq$ Probability — $p$-value: 0.156

Probability $\geq$ Harmonic Analysis — $p$-value: 0.113

Probability $\geq$ Combinatorics — $p$-value: 0.1049

Harmonic Analysis $\geq$ Combinatorics — $p$-value: 0.3824

Harmonic Analysis $\geq$ Algebra — $p$-value: 0.0961

Harmonic Analysis $\geq$ Algebraic Geometry — $p$-value: 0.1807

Combinatorics $\geq$ Algebra — $p$-value: 0.2181

Combinatorics $\geq$ Algebraic Geometry — $p$-value: 0.3265

Combinatorics $\geq$ Geometry — $p$-value: 0.1544

Combinatorics $\geq$ Analysis — $p$-value: 0.0719

Algebra $\geq$ Algebraic Geometry — $p$-value: 0.6461

Algebra $\geq$ Geometry — $p$-value: 0.3989

Algebra $\geq$ Dynamics — $p$-value: 0.0813

Algebra $\geq$ Mathematical Physics — $p$-value: 0.0546

Algebra $\geq$ Analysis — $p$-value: 0.1232

Algebraic Geometry $\geq$ Geometry — $p$-value: 0.2533

Algebraic Geometry $\geq$ Analysis — $p$-value: 0.0782

Geometry $\geq$ Number Theory — $p$-value: 0.0534

Geometry $\geq$ Dynamics — $p$-value: 0.1076

Geometry $\geq$ Mathematical Physics — $p$-value: 0.0704

Geometry $\geq$ Analysis — $p$-value: 0.1497

Number Theory $\geq$ Dynamics — $p$-value: 0.4906

Number Theory $\geq$ Mathematical Physics — $p$-value: 0.4514

Number Theory $\geq$ Analysis — $p$-value: 0.4417

Number Theory $\geq$ Applied Mathematics — $p$-value: 0.2717

Number Theory $\geq$ Group Theory — $p$-value: 0.2637

Number Theory $\geq$ Logic — $p$-value: 0.1557

Number Theory $\geq$ Complex Analysis — $p$-value: 0.0717

Dynamics $\geq$ Mathematical Physics — $p$-value: 0.4614

Dynamics $\geq$ Analysis — $p$-value: 0.4609

Dynamics $\geq$ Applied Mathematics — $p$-value: 0.3377

Dynamics $\geq$ Group Theory — $p$-value: 0.2999

Dynamics $\geq$ Logic — $p$-value: 0.1738

Dynamics $\geq$ Complex Analysis — $p$-value: 0.1272

Mathematical Physics $\geq$ Analysis — $p$-value: 0.4916

Mathematical Physics $\geq$ Applied Mathematics — $p$-value: 0.3506

Mathematical Physics $\geq$ Group Theory — $p$-value: 0.337

Mathematical Physics $\geq$ Logic — $p$-value: 0.23

Mathematical Physics $\geq$ Complex Analysis — $p$-value: 0.1423

Mathematical Physics $\geq$ History — $p$-value: 0.0525

Analysis $\geq$ Applied Mathematics — $p$-value: 0.4035

Analysis $\geq$ Group Theory — $p$-value: 0.3972

Analysis $\geq$ Logic — $p$-value: 0.3154

Analysis $\geq$ Complex Analysis — $p$-value: 0.2375

Analysis $\geq$ Lie Groups — $p$-value: 0.1041

Analysis $\geq$ History — $p$-value: 0.0881

Applied Mathematics $\geq$ Group Theory — $p$-value: 0.4655

Applied Mathematics $\geq$ Logic — $p$-value: 0.3609

Applied Mathematics $\geq$ Complex Analysis — $p$-value: 0.2362

Applied Mathematics $\geq$ Lie Groups — $p$-value: 0.0616

Applied Mathematics $\geq$ History — $p$-value: 0.0534

Group Theory $\geq$ Logic — $p$-value: 0.3728

Group Theory $\geq$ Complex Analysis — $p$-value: 0.2834

Group Theory $\geq$ Lie Groups — $p$-value: 0.0513

Logic $\geq$ Complex Analysis — $p$-value: 0.4276

Logic $\geq$ Lie Groups — $p$-value: 0.0613

Complex Analysis $\geq$ Lie Groups — $p$-value: 0.0871

Lie Groups $\geq$ History — $p$-value: 0.0531

Statistics $\geq$ History — $p$-value: 0.2517

History $\geq$ Other — $p$-value: 0.1533