Physical Sciences & Engineering
Mathematics research: funders, indexes, societies and standards
Mathematics research runs on shared scholarly infrastructure more than on laboratories or instruments. Work is classified with the Mathematics Subject Classification, MSC2020, which Mathematical Reviews and zbMATH maintain jointly and which has 63 two-digit, 529 three-digit and 6,022 five-digit classes. Publishers, reviewing services and funders use it to categorise papers and proposals. Preprints go to arXiv, whose math archive has 32 subject classes, and published work is indexed and reviewed in zbMATH Open, which has been free since January 2021, and in MathSciNet, the American Mathematical Society's online edition of Mathematical Reviews.
Funding comes from a short list of sources. The US National Science Foundation funds mathematical sciences through disciplinary programmes such as Algebra and Number Theory, Analysis, Combinatorics, Foundations, Topology and Mathematical Biology, plus Focused Research Groups, Research Training Groups and research institutes. The Simons Foundation supports Investigators, Collaborations and several mathematics institutes, and the Clay Mathematics Institute runs fellowships and the Millennium Prize Problems. Institutes such as SLMath in Berkeley, the Institute for Advanced Study and the Isaac Newton Institute host long residential programmes.
Learned societies set the venues of the field: the American, London and European Mathematical Societies, the Society for Industrial and Applied Mathematics, and the International Mathematical Union, which holds the quadrennial International Congress of Mathematicians and awards the Fields Medal. Researchers and administrators should know that papers are typeset in LaTeX, that author order in many mathematics papers is alphabetical rather than by contribution, and that shared references include the OEIS and the NIST Digital Library of Mathematical Functions. The specialization hubs below cover each branch of mathematics.
New to the field? What is Mathematics? →
Reviewed 5 November 2026 by CASRAI Editorial Board
- arXiv Mathematics archive(read 5 November 2026)
- NSF Mathematical Sciences(read 5 November 2026)
- MSC2020(read 5 November 2026)
- OEIS(read 5 November 2026)
- SIAM (Wikipedia)(read 5 November 2026)
- International Mathematical Union (Wikipedia)(read 5 November 2026)
- American Mathematical Society (Wikipedia)(read 5 November 2026)
- London Mathematical Society (Wikipedia)(read 5 November 2026)
- European Mathematical Society (Wikipedia)(read 5 November 2026)
- Simons Foundation (Wikipedia)(read 5 November 2026)
- Clay Mathematics Institute (Wikipedia)(read 5 November 2026)
- zbMATH Open (Wikipedia)(read 5 November 2026)
- Mathematical Reviews (Wikipedia)(read 5 November 2026)
- Mathematics Genealogy Project (Wikipedia)(read 5 November 2026)
- SLMath (MSRI) (Wikipedia)(read 5 November 2026)
- Digital Library of Mathematical Functions (Wikipedia)(read 5 November 2026)
- Annals of Mathematics (Wikipedia)(read 5 November 2026)
- LaTeX (Wikipedia)(read 5 November 2026)
Field at a glance
Journals
- Annals of Mathematics
- General mathematics journal published by Princeton University and the Institute for Advanced Study; founded 1874 as The Analyst, renamed in 1884.Website ↗
- Journal of the American Mathematical Society
- Quarterly peer-reviewed journal of the AMS covering pure and applied mathematics; first issue January 1988.Website ↗
- Journal of the European Mathematical Society
- Monthly peer-reviewed journal of the EMS covering pure and applied mathematics; founded 1999.Website ↗
Societies
- American Mathematical Society (AMS)
- Founded 1888 as the New York Mathematical Society; about 30,000 members, publishes the Bulletin and Notices, and hosts the Joint Mathematics Meetings.Website ↗
- London Mathematical Society
- Founded 1865; publishes the Proceedings, Journal and Bulletin of the LMS and awards the De Morgan Medal and research grants.Website ↗
- European Mathematical Society
- Founded 1990, seated in Helsinki; holds the European Congress of Mathematics every four years and runs EMS Press with over 25 journals.Website ↗
- Society for Industrial and Applied Mathematics (SIAM)
- Founded 1952; about 14,000 members, 19 research journals including SIAM Review, and about 23 activity groups.Website ↗
- International Mathematical Union (IMU)
- Reestablished in 1950; organises the International Congress of Mathematicians and awards the Fields, Abacus, Gauss, Chern and Leelavati prizes.Website ↗
Funders and programmes
- NSF Mathematical Sciences
- NSF programmes: eleven disciplinary programmes, Focused Research Groups, Research Training Groups, MATHINF and research institutes.Website ↗ · Our guide →
- Simons Foundation: Mathematics and Physical Sciences
- Funds Simons Investigators and Collaborations (launched 2012) and supports mathematics institutes including SLMath.Website ↗ · Our guide →
- Clay Mathematics Institute
- Founded 1998; offers Clay Research Fellowships and US$1,000,000 for each solved Millennium Prize Problem, of which one is solved.Website ↗
Archives, facilities and data
- On-Line Encyclopedia of Integer Sequences (OEIS)
- Searchable database of integer sequences kept by the OEIS Foundation; its home page showed 401,069 sequences on 5 November 2026.Website ↗
- NIST Digital Library of Mathematical Functions
- Online special-functions reference maintained by NIST since 2010, intended as an update of Abramowitz and Stegun.Website ↗
- Mathematics Genealogy Project
- Database of doctoral advisors and students hosted by North Dakota State University, with 334,029 people recorded as of September 2025.Website ↗
- Simons Laufer Mathematical Sciences Institute (SLMath)
- Berkeley institute founded 1982 as MSRI and renamed in 2022; runs semester programmes, workshops and summer research programmes.Website ↗
Standards and formats
- Mathematics Subject Classification 2020 (MSC2020)
- Joint Mathematical Reviews and zbMATH classification with 63 two-digit classes; free under CC BY-NC-SA as PDF, TeX or CSV.Website ↗
- LaTeX
- De facto standard for typesetting mathematical expressions, built on Knuth's TeX; amsmath is a standard package.Website ↗ · Our guide →
Preprints and indexes
- arXiv mathematics archive
- 32 math subject classes from Algebraic Geometry to Statistics Theory; math.IT aliases cs.IT and math.MP aliases math-ph.Website ↗ · Our guide →
- zbMATH Open
- Free since January 2021; indexes over 2,300 journals and more than 4 million records, with coverage from 1868.Website ↗
- Mathematical Reviews and MathSciNet
- AMS reviewing journal founded 1940 by Otto Neugebauer; MathSciNet is its online database, launched 1996.Website ↗
26 pages across 7 topic areas
Scholarly Publishing
- arXiv Math: Repository Guide for Mathematicians
Complete researcher guide to arXiv Mathematics: 32 subject classifications, TeX submission procedures, moderation policies, and dissemination norms.…
- ICERM Programs and Research Guide
ICERM research programs, semester fellowships, computational resources, and workshop application guidelines for mathematical sciences researchers.…
- IHES: Research Institute and Visiting Guide
A guide to the Institut des Hautes Études Scientifiques (IHES), covering visitor applications, research areas, funding, and publication outlets.…
- Math Genealogy Project: Academic Lineage Guide
A guide to the Mathematics Genealogy Project: tracing doctoral advisor trees, academic lineages, and intellectual history in the mathematical sciences…
- Mathematics Subject Classification (MSC)
An operational guide to the Mathematics Subject Classification (MSC2020), explaining code structure, author assignment rules, and mathematical indexin…
- NIST DLMF: Mathematical Functions Guide
A guide to the NIST Digital Library of Mathematical Functions (DLMF): formula search, equation permalinks, LaTeX export, and Abramowitz & Stegun linea…
- OEIS: Integer Sequences Research Guide
A practical guide to searching, citing, and submitting to the On-Line Encyclopedia of Integer Sequences (OEIS) for mathematical researchers.…
- Project Euclid: Math Publishing Platform
Guide to Project Euclid: governance by Cornell and Duke University Press, hosted mathematics journals, search features, open access, and platform comp…
- SLMath Institute Programs & Research Guide
SLMath research programs, visiting memberships, postdoctoral fellowships, and workshop application guidelines for mathematical sciences researchers.…
Laboratory Operations & Safety
- Calculus of Variations: Principles & Methods
A technical guide to the calculus of variations: Euler-Lagrange equations, boundary conditions, variational principles, and computational implementati…
- Ramsey Theory: Bounds, Principles, and Research
A comprehensive guide to Ramsey theory in combinatorics, covering exact Ramsey numbers, recent exponential bounds, and applications in computing resea…
- Riemannian Geometry: Core Concepts & Practice
A research-practice guide to Riemannian geometry: metric tensors, the Levi-Civita connection, curvature tensors, and applications in computational ana…
- Turing Patterns: Reaction-Diffusion Mechanism and Evidence
How Turing patterns arise from reaction and diffusion, where the mechanism is contested, and a five-point evidence checklist.…
Research Tools & Software
- BLAS (Basic Linear Algebra Subprograms): What It Is and How Research Software Uses It
BLAS is the low-level matrix library behind NumPy and R. Learn BLAS vs LAPACK, OpenBLAS vs MKL, and what to record for reproducibility.…
- FLINT (Fast Library for Number Theory): Install and First Program
How to install FLINT, the Fast Library for Number Theory, via C or python-flint, and verify a first computation against known output.…
- IEEE 754 Floating-Point Standard: Formats, Rounding and Reproducibility
What IEEE 754 defines, what it guarantees for reproducibility, and a checklist for reporting floating-point choices in research code.…
- MRLookup: AMS Citation Search Guide
How to use AMS MRLookup and Mref for Mathematical Reviews citation verification, BibTeX export, and free reference formatting.…
Grants Management & Research Funding
- Fields Medal: Rules, Criteria, and Program Guide
A comprehensive guide to the Fields Medal: IMU governance, under-40 eligibility criteria, selection procedures, and comparisons with major research pr…
- NSF MSPRF: Mathematical Sciences Postdoc Guide
Comprehensive guide to the NSF Mathematical Sciences Postdoctoral Research Fellowships (MSPRF): eligibility, tenure options, stipend rules, and applic…
- Simons Foundation Research Funding: Mathematics, Physical Sciences, Life Sciences, and Autism Research
How the Simons Foundation funds basic research across Mathematics and Physical Sciences, Life Sciences, SFARI autism research, and Neuroscience, and h…
Research Methods & Statistics
- Mathematics vs Statistics: What Is the Difference?Comparison
Mathematics proves results by deduction from assumptions; statistics draws conclusions from data under uncertainty. Side-by-side comparison of methods…
- What Is Control Theory and Systems Engineering Research?
Control theory is the engineering and mathematics of making systems behave as intended by measuring them and feeding corrections back. Methods, applic…
- What Is Mathematics? Research Areas, Funding, and Career Paths
A comprehensive guide to what mathematics is, its major subfields, the funding landscape for mathematics research (NSF DMS, DOE ASCR, NSA, private fou…
CRediT & Authorship
- Mathematics' Alphabetical Author Order: Origins and How It's Changing
Mathematics' alphabetical author-order convention traces to the Hardy-Littlewood collaboration and remains dominant, but adoption has been declining s…
- The Polymath Project and D.H.J. Polymath: How Crowdsourced Mathematics Decided Authorship
The Polymath Project, Timothy Gowers’s open blog-based mathematics collaboration, published results under the collective pseudonym D.H.J. Polymath rat…
Scholarly Writing
- Mathematics Paper Writing Conventions: Theorem-Proof Structure and Style
How mathematics research papers are structured around theorems, lemmas, proofs, and corollaries instead of IMRaD, plus the notation and AMS/LaTeX conv…
Key terms
- Acta Mathematica
- Advances in Mathematics
- Annals of Mathematics
- Compositio Mathematica: Profile & Metrics
- Crelle's Journal
- Duke Mathematical Journal
- Inventiones Mathematicae
- Journal of the American Mathematical Society
- Mathematische Annalen
- Notices of the AMS: Profile & Metrics
- Proceedings of the American Mathematical Society
- SIAM Review: Journal Impact & Metrics
