Mathematics research differs from research in most other academic disciplines in several fundamental ways. The primary form of evidence in mathematics is the proof — a logical argument demonstrating that a statement must be true given a set of axioms and previously established results. Mathematical research papers do not present data, conduct experiments, or test hypotheses in the way that empirical sciences do; they prove theorems, develop new mathematical structures, extend existing theories, or apply mathematical methods to problems in other fields. This guide, part of iResearchNet’s How to Do Research series, covers the databases, preprint servers, computational tools, and writing conventions that U.S. college and university students need for research papers in mathematics, applied mathematics, and statistics.
Academic Databases for Mathematics
MathSciNet
MathSciNet (Mathematical Reviews), maintained by the American Mathematical Society (AMS), is the discipline’s primary database and the standard tool for locating published mathematical research. MathSciNet indexes journal articles, books, and conference proceedings across all areas of pure and applied mathematics, and provides expert-written reviews of indexed publications — a feature that distinguishes it from most other academic databases. These reviews, written by mathematicians in the relevant subfield, summarize the results and methods of each paper and assess its contribution, providing an invaluable guide for students trying to determine whether a paper is relevant to their research before investing the time to read the full proof.
MathSciNet uses the Mathematics Subject Classification (MSC) system, a hierarchical classification scheme jointly maintained by the AMS and Zentralblatt MATH, to categorize research by subfield. The MSC organizes mathematics into 64 top-level areas (from 00-General to 97-Mathematics education), each subdivided into finer categories. A student researching problems in combinatorial graph theory, for example, can search under MSC code 05C (Graph theory) to find all indexed publications classified under that heading. Learning to use MSC codes is essential for efficient searching in mathematical databases.
Most U.S. university libraries provide access to MathSciNet through institutional subscription.
zbMATH Open
zbMATH Open (formerly Zentralblatt MATH), maintained by FIZ Karlsruhe in cooperation with the European Mathematical Society, is the other major database for mathematical research. It indexes a comparable body of literature to MathSciNet and also provides expert reviews. zbMATH has been transitioning to an open-access model, making an increasing portion of its content freely available. The two databases overlap substantially in journal coverage, but each includes some publications that the other does not, and the reviews are written by different reviewers — consulting both databases provides the most thorough coverage of the literature on a given topic.
arXiv
arXiv (maintained by Cornell University) is the primary preprint server for mathematics (and for physics, computer science, and several other quantitative fields). The vast majority of mathematics research papers are posted to arXiv before or simultaneously with submission to a peer-reviewed journal, making arXiv the fastest source of current mathematical research. Papers on arXiv have not undergone peer review, but because the culture of mathematics strongly values correctness (a proof is either valid or it is not), the quality of arXiv papers in mathematics is generally high — though errors do occur, and the absence of peer review means that readers must evaluate the arguments themselves.
arXiv papers are organized by subject area (math.AG for Algebraic Geometry, math.CO for Combinatorics, math.NT for Number Theory, and so on), and students can browse recent submissions within their area of interest or search across the entire archive by keyword, author, or title. For students working on topics at the frontier of current research, arXiv is indispensable because journal publication in mathematics typically takes one to three years, meaning that papers appearing in journals today were often completed two or more years before publication.
Additional Resources
Google Scholar provides broad indexing and citation tracking across mathematics and is useful for identifying how a specific paper has been cited and extended by subsequent work. JSTOR archives complete runs of several major mathematics journals. The Digital Mathematics Library (DML) is a coordinated effort to digitize the mathematical literature, and several national projects — including the Numdam project in France and Project Euclid — provide free access to digitized mathematical journals, many of which contain classical papers that remain relevant to current research.
Types of Mathematics Research Papers
Students encounter several distinct types of mathematical writing at U.S. colleges and universities, and the type of paper determines the research methods and sources required.
Proof-Based Papers
At the graduate level and in advanced undergraduate courses, mathematics research papers present original proofs — logically rigorous arguments establishing that specific mathematical statements are true. A proof-based paper typically states a theorem, proves it, and discusses its relationship to existing results. This type of paper requires the deepest engagement with the mathematical literature, because the student must demonstrate familiarity with related theorems, understand the techniques used in previous proofs, and position the new result within the existing body of knowledge.
Writing a proof-based paper requires mathematical maturity that develops over years of study. Undergraduate students are more likely to encounter proof-based writing in homework assignments and examinations than in research papers, though senior theses and honors projects in mathematics departments frequently involve original or expositional proofs.
Expository Papers
Expository papers explain existing mathematics — a theorem, a proof technique, a subfield, or a historical development — to a specific audience without necessarily presenting new results. Many undergraduate mathematics research papers are expository: the student selects a mathematical topic, reads the relevant literature, and writes a clear, accurate account that synthesizes and explains the key ideas. A strong expository paper does more than summarize; it organizes the material in a way that illuminates connections, provides motivation for definitions and constructions that might otherwise seem arbitrary, and helps the reader understand not just what is true but why it is true and why it matters.
Applied Mathematics Papers
Applied mathematics papers use mathematical methods — modeling, simulation, optimization, statistical analysis — to address problems arising in science, engineering, economics, or other fields. These papers require both mathematical skill and domain knowledge, and the literature search may need to span multiple databases (MathSciNet for the mathematical methods, and a discipline-specific database for the application domain). Applied mathematics papers often include computational results alongside analytical arguments, and the writing must explain both the mathematical framework and its application clearly enough for readers in the target field.
History of Mathematics Papers
Papers on the history of mathematics examine the development of mathematical ideas, the lives and contributions of mathematicians, or the cultural and institutional contexts in which mathematical knowledge has been produced. These papers draw on primary sources (original publications, letters, notebooks) and secondary historical scholarship, using methods closer to historical research than to mathematical proof. Students writing history of mathematics papers should consult MathSciNet (which indexes historical scholarship on mathematics), JSTOR (which archives journals in the history of science), and the MacTutor History of Mathematics Archive (University of St Andrews), which provides biographies, historical articles, and timelines covering the development of mathematical ideas from antiquity to the present.
Mathematical Writing and Typesetting
LaTeX
The standard typesetting system for mathematical writing is LaTeX, a document preparation system that produces professionally formatted mathematical notation, equations, diagrams, and bibliographies. Virtually all mathematics journals require submissions in LaTeX format, and most mathematics departments at U.S. universities expect students in upper-division and graduate courses to write papers and problem sets in LaTeX. Learning LaTeX is a non-trivial investment of time — the system uses a markup language rather than a WYSIWYG interface — but the investment pays dividends across every subsequent mathematics course and in any career that involves technical writing.
Overleaf provides a free, browser-based LaTeX editor with real-time collaboration, templates for common document types (including AMS journal article templates), and a rich library of tutorials. Most students find Overleaf the easiest way to begin using LaTeX. Desktop LaTeX distributions — TeX Live (cross-platform), MiKTeX (Windows), and MacTeX (macOS) — provide offline alternatives for students who prefer local installations.
Writing Conventions
Mathematical writing follows conventions that differ from writing in other disciplines. Theorems, lemmas, propositions, and corollaries are stated formally and set apart from the surrounding text. Proofs begin with “Proof.” and end with a symbol (typically a square or QED). Definitions are precise and unambiguous — every term must be defined before it is used, and the meaning of a defined term is exactly what the definition says, no more and no less.
The prose surrounding formal mathematical statements should be clear, concise, and logically ordered. Contrary to a common misconception, good mathematical writing is not a sequence of formulas with minimal connecting text; it is a carefully constructed argument in which English sentences explain the logical flow, motivate the steps, and help the reader understand why each step follows from the previous one. The AMS publishes a style guide, and several widely read essays on mathematical writing — including those by Paul Halmos, Steven Krantz, and the AMS — provide detailed guidance that students should consult.
Computational Tools
Modern mathematical research increasingly involves computational work — symbolic computation, numerical simulation, data visualization, and experimental mathematics (using computation to discover patterns and formulate conjectures before attempting formal proofs).
Mathematica (Wolfram Research) and Maple (Maplesoft) are the two major commercial computer algebra systems used in mathematics departments. Both can perform symbolic computation (algebraic manipulation, calculus, differential equations), numerical computation, and visualization. Many U.S. universities provide site licenses for one or both.
SageMath is a free, open-source mathematics software system that integrates many existing open-source packages (NumPy, SciPy, Matplotlib, R, and others) under a unified Python-based interface. MATLAB (MathWorks) is widely used for numerical computation and simulation, particularly in applied mathematics and engineering. R and Python (with libraries such as NumPy, SciPy, SymPy, and Matplotlib) are increasingly used for statistical analysis, data visualization, and computational mathematics.
GeoGebra provides free, interactive geometry, algebra, and calculus tools that are particularly useful for visualizing mathematical objects and for generating figures for mathematical papers.
Key Journals in Mathematics
The leading general-interest mathematics journals include the Annals of Mathematics (Princeton), Journal of the American Mathematical Society (JAMS), Inventiones Mathematicae, Acta Mathematica, and the Duke Mathematical Journal. These journals publish research across all areas of pure mathematics and are among the most selective in any academic field.
Subfield journals serve specific areas. Journal of Algebra, Transactions of the American Mathematical Society, Advances in Mathematics, Combinatorica, Journal of Number Theory, Topology, SIAM Journal on Applied Mathematics, SIAM Review, and the Annals of Statistics are among the many journals that publish research within particular mathematical subfields. For applied mathematics and interdisciplinary work, SIAM (the Society for Industrial and Applied Mathematics) publishes a suite of journals covering numerical analysis, optimization, scientific computing, and mathematical modeling.
For expository mathematics accessible to advanced undergraduates, the American Mathematical Monthly (published by the Mathematical Association of America), Mathematics Magazine, and The College Mathematics Journal publish articles, problems, and reviews aimed at a broad mathematical audience rather than specialists in a single subfield.
Professional Associations
The American Mathematical Society (AMS) is the primary professional association for research mathematicians in the United States. It publishes major journals, maintains MathSciNet, and hosts the Joint Mathematics Meetings (JMM) — the largest annual gathering of mathematicians in the world. The Mathematical Association of America (MAA) focuses on mathematics education and collegiate mathematics, publishing the American Mathematical Monthly and hosting conferences and competitions for undergraduate students. The Society for Industrial and Applied Mathematics (SIAM) serves applied mathematicians, computational scientists, and engineers.
Both the AMS and the MAA offer student membership at reduced rates, including access to journals, conference discounts, and career resources. Undergraduate students interested in mathematical research should consider participating in MAA MathFest, the JMM undergraduate poster session, and the Research Experiences for Undergraduates (REU) programs funded by the National Science Foundation (NSF), which provide intensive summer research opportunities at universities across the United States.
Reading and Understanding Mathematical Papers
Reading a mathematics research paper is a qualitatively different activity from reading a paper in most other disciplines, and students new to mathematical literature should expect the process to be slow, nonlinear, and effortful. A ten-page mathematics paper may require ten hours or more to read carefully, because every statement must be verified — the reader should check that each step of a proof follows logically from the previous steps and from the cited results.
Effective strategies for reading mathematical papers include reading the abstract and introduction first to understand the main result and its context, then reading the statement of the main theorem to understand precisely what is being claimed, then skimming the proof structure (the major lemmas and their logical relationships) before attempting to follow the detailed arguments. Students should keep scratch paper available to work through calculations, construct examples, and test their understanding of definitions by checking whether specific objects satisfy or fail to satisfy the conditions.
When a paper cites a result from another publication, the reader faces a choice: take the cited result on faith and continue, or locate the cited paper and verify the result independently. At the undergraduate level, taking cited results on faith is usually reasonable — attempting to verify every cited result in a paper that draws on decades of prior work is impractical. However, the student should understand the statement of each cited result clearly enough to see why it applies in the context where it is being used.
Mathematical notation can be a significant barrier for students encountering a subfield for the first time, because different areas of mathematics use different notational conventions, and the same symbol may mean different things in different contexts. Keeping a running glossary of notation — recording each symbol’s meaning when it is first defined — prevents confusion when the symbol recurs pages later in a complex argument.
Mathematics Education Research
A distinct but related field of scholarship examines how mathematics is taught, learned, and understood. Mathematics education research — published in journals such as the Journal for Research in Mathematics Education (JRME, published by the National Council of Teachers of Mathematics), Educational Studies in Mathematics, Mathematical Thinking and Learning, and For the Learning of Mathematics — applies methods from education research and cognitive psychology to questions about mathematical learning.
Students writing research papers on topics such as how students learn algebra, why certain mathematical concepts are persistently difficult, how technology affects mathematics instruction, or how different curricula compare in effectiveness are conducting mathematics education research rather than mathematical research. This distinction matters because the databases, methods, and writing conventions differ. Mathematics education papers are indexed in ERIC and PsycINFO (in addition to MathSciNet, which indexes some education-oriented mathematics journals), and they follow the methodological conventions of social science research — research questions, literature reviews, data collection and analysis, and discussion of findings — rather than the theorem-proof structure of mathematical papers.
The National Council of Teachers of Mathematics (NCTM) publishes curriculum standards, position statements, and practitioner-oriented journals (Mathematics Teacher: Learning and Teaching PK-12) alongside the research-oriented JRME. The Conference Board of the Mathematical Sciences (CBMS) publishes reports on undergraduate mathematics education, including surveys of enrollment, curriculum, and pedagogy at U.S. colleges and universities. Students writing on mathematics education topics should consult these organizations’ publications in addition to the academic journal literature.
Data Sources for Applied Mathematics and Statistics
Students writing applied mathematics or statistics papers may need access to datasets for analysis, modeling, or illustration. Several repositories provide freely accessible data:
UCI Machine Learning Repository provides standard datasets widely used in statistics and machine learning research. Kaggle hosts datasets and competitions covering a wide range of applied statistical problems. The Federal Reserve Economic Data (FRED) provides time-series economic data suitable for time-series analysis and econometric modeling. The Census Bureau provides demographic data at multiple geographic scales. The Inter-university Consortium for Political and Social Research (ICPSR) archives social science datasets suitable for statistical analysis.
For computational mathematics and numerical methods, standard test problems and benchmark datasets are published in journals such as ACM Transactions on Mathematical Software and are maintained by organizations like the National Institute of Standards and Technology (NIST), which publishes reference datasets and mathematical function libraries (the NIST Digital Library of Mathematical Functions).
Frequently Asked Questions
What is the best database for mathematics research?
MathSciNet is the field’s primary database and should be the starting point for any mathematics research project. For the most current research (papers not yet published in journals), arXiv provides preprint access. zbMATH Open complements MathSciNet with additional coverage and is increasingly available in open-access form. Google Scholar provides useful citation tracking for identifying how a specific result has been used and extended.
Do I need to know LaTeX to write a math paper?
For advanced undergraduate and graduate mathematics courses, LaTeX is expected — mathematical notation cannot be rendered adequately in standard word processors, and virtually all mathematics journals require LaTeX-formatted submissions. For introductory courses, instructors may accept handwritten or word-processed assignments. Students who anticipate taking upper-division mathematics courses should learn LaTeX early; Overleaf provides the gentlest learning curve.
How do I find papers on a specific mathematical topic?
Start with a MathSciNet search using the relevant MSC code combined with keyword terms. Read the expert reviews to identify the most relevant papers. Follow references backward (reading the papers that your source cites) and forward (using MathSciNet’s or Google Scholar’s citation tracking to find later papers that cite your source). For survey articles that synthesize the state of knowledge in a subfield, check the Bulletin of the American Mathematical Society and the Notices of the American Mathematical Society, both of which publish accessible overviews of active research areas.
Can I write an expository paper rather than proving new theorems?
Many undergraduate mathematics courses — and some graduate courses — assign expository papers that explain existing mathematics to a specific audience. A strong expository paper demonstrates deep understanding of the material, presents it clearly and accurately, provides motivation and intuition alongside formal statements, and cites the original sources appropriately. Expository writing is a valued skill in mathematics; some of the most cited and influential mathematical publications are survey and expository articles rather than papers presenting new results.
Related Pages
- How to Do Research — Overview of research methods and discipline-specific guides
- How to Do Research on Science and Technology — Sources and methods for STEM research
- How to Do Library Research — Using catalogs, databases, and librarian expertise
- How to Do Online Research — Evaluating and using internet sources
- How to Write a Research Paper — Complete step-by-step guide