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Subfield · Mathematics

Mathematical Physics research

Mathematical Physics is a subfield of Mathematics. Science Explorer counts 257k research works in it since 1950. 10.0% of them reached the world's top 10% most cited for their field and year.

In short: use of mathematics to solve physics problems.

Research works
257k
fractional, since 1950
In the world top 10%
26k
per year above
Top-10% rate
10.0%
share of its works in the world top 10%
Topics
12
Growth, 2018-21 → 2022-25
+17%
the tick is no change
Largest producer
China
26% of the 10 largest, 2022-now
Mathematical Physics

What is Mathematical Physics research about, and what is growing?

By the world's output in 2021-25, its most used keywords are Ergodic Theory, K-Theory, Inverse Problems, Regularization Methods, Global Well-Posedness, Nonlinear Schrödinger Equation. Gaining share since 2011-15: Global Well-Posedness, Nonlinear Schrödinger Equation, p-adic, ultrametric, Mathematical Physics. Losing share: Interacting Particle Systems, Scaling Limits.

▲ gaining share of this field◆ holding its share▼ losing share

Size is the world's fractional works in 2021-25 in the topics tagged with each word; colour compares the word's share of this field with 2011-15. Each links to the topic it comes from most.

All 28 words, with their numbers
  1. Ergodic Theory7,771◆ 1.01×2 topics
  2. K-Theory6,556◆ 0.93×2 topics
  3. Inverse Problems5,697◆ 1.07×1 topic
  4. Regularization Methods5,697◆ 1.07×1 topic
  5. Global Well-Posedness5,598▲ 1.31×1 topic
  6. Nonlinear Schrödinger Equation5,598▲ 1.31×1 topic
  7. Chaos4,752◆ 0.91×1 topic
  8. Dynamical Systems4,752◆ 0.91×1 topic
  9. Automorphic Forms4,664◆ 1.02×1 topic
  10. Langlands Conjectures4,664◆ 1.02×1 topic
  11. Deformation Quantization4,357◆ 0.95×1 topic
  12. Poisson Manifolds4,357◆ 0.95×1 topic
  13. Mathematical Physics3,891▲ 1.18×2 topics
  14. Nonlinear Equations3,891▲ 1.18×2 topics
  15. Differential Operators3,700◆ 1.02×1 topic
  16. Spectral Theory3,700◆ 1.02×1 topic
  17. Interacting Particle Systems3,372▼ 0.77×1 topic
  18. Scaling Limits3,372▼ 0.77×1 topic
  19. p-adic3,020▲ 1.21×1 topic
  20. ultrametric3,020▲ 1.21×1 topic
  21. Banach Spaces2,403◆ 0.90×1 topic
  22. Operators2,403◆ 0.90×1 topic
  23. C*-Algebras2,199◆ 0.89×1 topic
  24. Quantum Groups2,199◆ 0.89×1 topic
  25. Generalized Functions1,545◆ 0.89×1 topic
  26. Infinite and Infinitesimal Numbers1,545◆ 0.89×1 topic
  27. Neutrosophy871◆ 1.07×1 topic
  28. Smarandache Function871◆ 1.07×1 topic

Which countries lead Mathematical Physics research?

On research quality in 2022-now, the United States, China and Germany rank highest in Mathematical Physics. By volume, China and the United States publish the most (6.7k and 5.3k works in 2022-now).

By research quality, 2022-now

  1. 1 United States #1 all time 1.19× citation excellence · 1.52× journals · 100% breadth 11.8% top 10% · 4.7k works
  2. 2 China #4 all time 0.97× citation excellence · 0.97× journals · 100% breadth 10.8% top 10% · 6k works
  3. 3 Germany #3 all time 1.24× citation excellence · 1.40× journals · 100% breadth 11.7% top 10% · 1.5k works
  4. 4 United Kingdom #5 all time 1.33× citation excellence · 1.46× journals · 100% breadth 13.1% top 10% · 1.2k works
  5. 5 France #2 all time 1.00× citation excellence · 1.25× journals · 100% breadth 8.7% top 10% · 2.2k works
  6. 6 Switzerland #6 all time 1.55× citation excellence · 1.87× journals · 100% breadth 15.1% top 10% · 294 works
  7. 7 Italy #7 all time 1.35× citation excellence · 0.70× journals · 100% breadth 12.2% top 10% · 1.2k works
  8. 8 Canada #9 all time 1.00× citation excellence · 1.19× journals · 100% breadth 10.3% top 10% · 639 works
  9. 9 Spain #10 all time 1.13× citation excellence · 1.00× journals · 100% breadth 11.5% top 10% · 710 works
  10. 10 Sweden #15 all time 1.29× citation excellence · 1.39× journals · 100% breadth 12.3% top 10% · 255 works

The track on each row is that row's top-10% rate. The ranking itself is the quality score, which compares the rate with what each place's own topics predict, adds its share of papers in the Nature Index journals, and counts output and breadth too.

Of 32 countries ranked in Mathematical Physics. Ranked on the quality score for 2022-now; the bracket is the all-time rank.

All countries ranked in Mathematical Physics

By volume, 2022-now

  1. 1 China 6.7k works
  2. 2 United States 5.3k works
  3. 3 Russia 2.5k works
  4. 4 France 2.4k works
  5. 5 India 1.7k works
  6. 6 Germany 1.7k works
  7. 7 Japan 1.5k works
  8. 8 Italy 1.4k works
  9. 9 United Kingdom 1.3k works
  10. 10 Brazil 930 works

How concentrated that is

The same countries as shares of everything the list above accounts for. A node where two countries do two thirds of the work and one spread evenly across twelve read alike as a ranking and not at all alike here.

China: 26.2%United States: 20.9%Russia: 9.6%France: 9.3%6 others listed: 33.9%26%largest
China6,681 · 26.2%United States5,318 · 20.9%Russia2,450 · 9.6%France2,375 · 9.3%6 others listed8,640 · 33.9%

Shares of the rows listed above, not of the whole node.

Which institutions lead Mathematical Physics research?

Centre National de la Recherche Scientifique (France) ranks first in Mathematical Physics on research quality in 2022-now, followed by ETH Zurich and Massachusetts Institute of Technology.

By research quality, 2022-now

  1. 1 Centre National de la Recherche Scientifique France · Paris · #1 all time 8.5% top 10% · 1.1k works
  2. 2 ETH Zurich Switzerland · Zurich · #3 all time 17.6% top 10% · 84 works
  3. 3 Massachusetts Institute of Technology United States · Cambridge · #4 all time 18.2% top 10% · 100 works
  4. 4 University of Oxford United Kingdom · Oxford · #15 all time 16.7% top 10% · 113 works
  5. 5 Max Planck Society Germany · Munich · #11 all time 19.2% top 10% · 108 works
  6. 6 Princeton University United States · Princeton · #2 all time 22.1% top 10% · 65 works
  7. 7 University of California, Berkeley United States · Berkeley · #7 all time 17.7% top 10% · 78 works
  8. 8 University of Cambridge United Kingdom · Cambridge · #10 all time 14.9% top 10% · 85 works
  9. 9 New York University United States · New York · #8 all time 18.0% top 10% · 81 works
  10. 10 Columbia University United States · New York · #16 all time 18.2% top 10% · 65 works

The track on each row is that row's top-10% rate. The ranking itself is the quality score, which compares the rate with what each place's own topics predict, adds its share of papers in the Nature Index journals, and counts output and breadth too.

Of 63 institutions above the floor of 40 fractional works in Mathematical Physics.

Top 100 Mathematical Physics institutions

By volume, 2022-now

  1. 1 Centre National de la Recherche ScientifiqueFrance 1.2k works
  2. 2 Russian Academy of SciencesRussia 719 works
  3. 3 Lomonosov Moscow State UniversityRussia 269 works
  4. 4 Chinese Academy of SciencesChina 179 works
  5. 5 National Research University Higher School of EconomicsRussia 156 works
  6. 6 Tsinghua UniversityChina 153 works
  7. 7 Polish Academy of SciencesPoland 140 works
  8. 8 St Petersburg UniversityRussia 136 works
  9. 9 Kyoto UniversityJapan 134 works
  10. 10 University of OxfordUnited Kingdom 123 works

Who are the leading researchers in Mathematical Physics?

Counting only their work in Mathematical Physics, Dachun Yang, Barry Simon and Jean Bourgain have the most papers in the world top 10% for their subfield and year beyond the 10% expected.

  1. 1 Dachun Yang China · #30,393 worldwide overall 57.1 of 145 papers here in the top 10%
  2. 2 Barry Simon United States · #13,092 worldwide overall 50.7 of 128 papers here in the top 10%
  3. 3 Jean Bourgain United States · #9,175 worldwide overall 43.7 of 99 papers here in the top 10%
  4. 4 Gunther A. Uhlmann United States · #37,489 worldwide overall 37.3 of 85 papers here in the top 10%
  5. 5 Satya N. Majumdar France · #35,545 worldwide overall 35.2 of 69 papers here in the top 10%
  6. 6 Carlos E. Kenig United States · #18,685 worldwide overall 37.6 of 96 papers here in the top 10%
  7. 7 Terence Tao United States · #6,318 worldwide overall 34.2 of 62 papers here in the top 10%
  8. 8 Frank Merle France · #16,627 worldwide overall 30.2 of 56 papers here in the top 10%
  9. 9 Daniel Tataru United States · #29,720 worldwide overall 27.0 of 54 papers here in the top 10%
  10. 10 James A. Yorke United States · #11,109 worldwide overall 28.2 of 69 papers here in the top 10%

Ranked on their work in Mathematical Physics alone: how many more of their papers here reached the world top 10% for their subfield and year than the 10% expected, among 247,715 ranked researchers with a paper here. Each paper counts once, so many strong papers outrank one famous one, and a long record of ordinary papers does not climb. Identities come from OpenAlex author disambiguation, which occasionally merges or splits people.

Top 100 Mathematical Physics researchers

What are the most cited papers in Mathematical Physics?

The most cited paper in Mathematical Physics is "CONFIDENCE LIMITS ON PHYLOGENIES: AN APPROACH USING THE BOOTSTRAP" (Joseph Felsenstein, 1985), with 41,763 citations. Next come "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems" (12,347) and "Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models" (8,374).

Most cited papers
#PaperReleasedCitations
1 CONFIDENCE LIMITS ON PHYLOGENIES: AN APPROACH USING THE BOOTSTRAP Joseph Felsenstein · Evolution Open access Jul 1985 41,763
2 A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems Amir Beck, Marc Teboulle · SIAM Journal on Imaging Sciences 2009 12,347
3 Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models N. David Mermin, Hermann‐Josef Wagner · Physical Review Letters 28 Nov 1966 8,374
4 On the generators of quantum dynamical semigroups Göran Lindblad · Communications in Mathematical Physics Jun 1976 7,738
5 Measuring the strangeness of strange attractors Peter Grassberger, Itamar Procaccia · Physica D Nonlinear Phenomena Oct 1983 5,689
All 100 most cited papers, with filters and sorting

Where is Mathematical Physics research done?

The largest centres of Mathematical Physics research in 2022-now are Moscow (Russia), Beijing (China), Paris (France) and Tokyo (Japan). Among places with at least 20 works in it, it is an unusually large share of all research in Bures-sur-Yvette, Huairou and Vladikavkaz.

Largest cities, 2022-now

  1. 1 MoscowRussia 1.2k works
  2. 2 BeijingChina 1k works
  3. 3 ParisFrance 795 works
  4. 4 TokyoJapan 493 works
  5. 5 ShanghaiChina 488 works
  6. 6 NanjingChina 371 works
  7. 7 GuangzhouChina 287 works
  8. 8 LondonUnited Kingdom 281 works
  9. 9 New YorkUnited States 275 works
  10. 10 WuhanChina 256 works

Where it is the local speciality

  1. Bures-sur-YvetteFR · 26.0 works56×
  2. Huairou20.8 works51×
  3. VladikavkazRU · 48.8 works24×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Mathematical Physics than the world average.

See Mathematical Physics on the map

Mathematical Physics research by country

The best places for Mathematical Physics research inside each country, with that country's researchers and world standing.

Where is the best place to study Mathematical Physics?

Among universities, judged by research, Princeton University, ETH Zurich and Massachusetts Institute of Technology score highest, combining standing in the field's ranking, citation excellence, specialisation, size, growth and international reach. Research strength is one signal when choosing where to study; it does not measure teaching.

0%10%20%mean 17.77%fractional works in this field (log) →share of work in the world top 10% →Princeton UniversityMassachusetts Instit…University of Califo…ETH Zurich
above the meannear itbelow it

One dot per university in the first ten of the table below, at the default weights. The upper left is the interesting corner: small places doing unusually strong work.

Standingthis field's institution ranking, three quarters all time and a quarter 2022-now
50%
Citation excellenceshare of its work in the world top 10%, 2022-now and all time averaged
20%
Specialisationhow much of its work is in this field, counted up to 10× the world share
5%
Sizeits output in this field, on a log scale
15%
Growthoutput growth, counted from -20% to +20%
5%
Internationalshare of its work with a co-author abroad
5%
Each slider sets a weight; the share it gets of the score is beside it.
#UniversityScoreRank in field (all time)Top 10%SpecialisationWorksGrowth
1Princeton UniversityUnited States 85.4#6 (#2)21.4%5.1×73 +10.5%
2ETH ZurichSwitzerland 83.2#2 (#3)17.0%4.2×95 +3.4%
3Massachusetts Institute of TechnologyUnited States 82.1#3 (#4)17.9%4.8×110 +11.2%
4University of California, BerkeleyUnited States 72.3#7 (#7)17.8%2.7×84 +17.1%
5New York UniversityUnited States 71.4#9 (#8)16.6%2.5×94 +22.1%
6University of OxfordUnited Kingdom 69.6#4 (#15)16.4%2.7×123 +26.4%
7University of CambridgeUnited Kingdom 65.0#8 (#10)14.5%2.4×93 -7.5%
8Columbia UniversityUnited States 65.0#10 (#16)18.6%2.0×70 +35.0%
9Harvard UniversityUnited States 64.9- (#6)20.0%0.4×40 +16.0%
10Stanford UniversityUnited States 64.6- (#5)17.5%1.3×61 +33.0%

Universities only. The score blends standing (50%), citation excellence (20%), size (15%), specialisation (5%), growth (5%), international reach (5%), 2022-now; growth compares 2018-21 with 2022-25 and counts up to +20%, since a jump from a small base is mostly noise. Standing is the score in this field's institution ranking, three quarters over all time and a quarter 2022-now; excellence averages the two windows too. Four years alone undercount fields such as AI, where many conference and arXiv papers from 2022 on carry no affiliation in the data.

Every institution in Mathematical Physics, country by country

Is Mathematical Physics research growing?

Output in 2022-2025 was 17% higher than in 2018-2021, peaking in 2024. The fastest-growing topics are advanced mathematical theories, Advanced Algebra and Geometry and Advanced Mathematical Physics Problems.

still counting19801990200020102020
grewheldshrank

The same series as a ribbon: one cell per year, darker for more. The line above answers how much; this answers when.

Its share of all research

Growing output can still be a shrinking share: all of science grows. This is Mathematical Physics as a percentage of every work published that year, 0.16% in 2024.

still counting0.20%0.40%0.60%Mathematical Physics0.15% in 202519801990200020102020

Fractional works of Mathematical Physics over all fractional works that year. How the balance of science has shifted.

How much of it reaches the world top 10%

Year by year.

still counting5%10%15%Mathematical Physics10.2% in 2025200020102020

Of each year's works in it, the share that reached the world's 10% most cited for their field and year. A field is the world for its own topics, so there is no other line to hold it against.

Which topics inside it are moving

Growth and decline on one axis around a shared zero. Two lists side by side hide the thing that matters: whether the growth dwarfs the decline, or the other way round.

  1. advanced mathematical theories+27%
  2. Advanced Algebra and Geometry+26%
  3. Advanced Mathematical Physics Problems+23%
  4. Numerical methods in inverse problems+19%
  5. Spectral Theory in Mathematical Physics+17%
  6. Advanced Mathematical Theories+17%
← shrinkingno changegrowing →

Growth in output from 2018-21 to 2022-25. Every topic is a page of its own.

What does Mathematical Physics include?

Mathematical Dynamics an…31kMathematical Dynamics and Fractals: 31k fractional worksNumerical methods in inv…31kNumerical methods in inverse problems: 31k fractional worksHomotopy and Cohomology in Algebraic Topology: 28k fractional worksAdvanced Algebra and Geo…27kAdvanced Algebra and Geometry: 27k fractional worksStochastic processes and…26kStochastic processes and statistical mechanics: 26k fractional worksSpectral Theory in Mathematical Physics: 23k fractional worksAdvanced Mathematical Physics Problems: 22k fractional worksadvanced mathematical th…18kadvanced mathematical theories: 18k fractional worksAdvanced Banach Space Theory: 18k fractional worksAdvanced Operator Algebra Research: 15k fractional worksMathematical and Theoretical Analysis: 12k fractional worksAdvanced Mathematical Theories: 5,642 fractional works

Area is fractional works in that topic. Every name below is a page of its own.

Share who leads Mathematical Physics

Who leads Mathematical Physics, on Science Explorer

A card with the countries and institutions that lead Mathematical Physics, for a post or a slide.

Download the card

1200 × 630 PNG, sized for social posts, drawn from the latest snapshot.

The leading institutions' badges6Each links to that institution's page and updates with every snapshot

Discussion

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