What Is Mathematics? Research Areas, Funding, and Career Paths

Mathematics is the formal science that studies quantity, structure, space, and change through abstract reasoning and logical proof rather than empirical observation. Where a natural scientist tests a claim against data collected from the physical world, a mathematician establishes a claim by deriving it, through a chain of valid logical steps, from a small set of accepted starting assumptions (axioms) and previously proven results. That reliance on proof rather than observation is what places mathematics — alongside logic, theoretical computer science, and statistics — in the formal sciences, distinct from the natural sciences (which test theories against the physical world) and the social sciences (which test theories against human behavior and society).

This guide answers “what is mathematics” in real depth: its core questions and methods, its major subfields, and — because this page is published by CASRAI, a research-administration standards body — the funding landscape, research methods, and career pathways that matter to anyone conducting or administering mathematics research specifically. It is part of CASRAI’s Branches of Science series.

What Is Mathematics? A Working Definition

Mathematics is conventionally divided into two broad, overlapping modes of practice. Pure mathematics studies mathematical structures and their properties for their own sake — numbers, shapes, functions, symmetries, and the logical relationships between them — without requiring an immediate application outside mathematics itself. Applied mathematics develops and uses mathematical tools and models to solve problems that originate in other fields: physics, engineering, biology, economics, computer science, and beyond. The boundary between the two is porous and shifts over time — results discovered as pure mathematics (number theory is the standard example) have repeatedly turned out, decades later, to be essential to applied fields such as cryptography.

What unifies both modes is the axiomatic method: mathematics builds outward from a small set of explicitly stated assumptions, using deductive logic to prove theorems that follow necessarily from those assumptions. A mathematical claim is not considered established because it has been checked against many examples or because most experts believe it — it is established only once a valid proof exists, or it remains an open conjecture. This is a fundamentally different standard of evidence than the natural sciences use, and it is why mathematical results, once proven correct, do not get revised the way scientific theories do as new data arrives — a correct proof of the Pythagorean theorem is exactly as true today as it was in antiquity.

Mathematical research proceeds through a recognizable cycle: identifying an interesting question or pattern, formulating a precise conjecture about it, attempting to prove or disprove that conjecture, and — once a proof exists — submitting it to peer review and publication so the broader community can verify the reasoning. Some of the field’s best-known open problems illustrate how long this cycle can run: the Clay Mathematics Institute’s seven Millennium Prize Problems, each carrying a $1 million prize, were named in 2000; as of this writing only one, the Poincaré conjecture, has been solved (by Grigori Perelman, who declined both the prize money and the Fields Medal he was also awarded for the work).

How Mathematics Relates to Neighboring Disciplines

Mathematics sits at the foundation of most other formal, natural, and applied sciences, and its relationship to several neighboring fields is close enough to be worth spelling out:

• Physics. Mathematics and physics have developed hand-in-hand since Newton and Leibniz independently invented calculus specifically to describe motion and change; large parts of modern mathematics (differential geometry, functional analysis, group representation theory) were developed by, or in close collaboration with, physicists working on problems in mechanics, electromagnetism, and quantum theory. See CASRAI’s guide to physics.
• Statistics. Statistics grew out of probability theory, itself a branch of mathematics, and remains grounded in it — but statistics is widely treated as a distinct discipline because so much of its content (study design, applied data analysis, domain-specific methodology) is not purely mathematical. See CASRAI’s guide to statistics for the fuller distinction.
• Computer science. Theoretical computer science — algorithms, computational complexity, cryptography, formal logic, discrete mathematics — is essentially a branch of applied mathematics, and the two fields share departments, conferences, and researchers at many universities. See CASRAI’s guide to computer science.
• Economics. Modern economics relies heavily on mathematical modeling — optimization, game theory, differential equations, and statistical/econometric methods — to formalize theories of markets, decision-making, and growth. See CASRAI’s guide to economics.
• Artificial intelligence. Machine learning methods are built directly on mathematical foundations — linear algebra (representing data and model weights), calculus and optimization theory (training via gradient-based methods), and probability theory (uncertainty and generative modeling). See CASRAI’s guide to artificial intelligence.
• Engineering. Applied mathematics — differential equations, linear algebra, numerical methods, optimization — is the common mathematical toolkit underlying essentially every engineering discipline’s quantitative work, from structural analysis to control systems. CASRAI’s guide to industrial engineering covers one field where applied mathematics (operations research, optimization) is especially central.

Major Subfields and Branches of Mathematics

Mathematics is commonly organized into the following broad areas, though most active research sits at the intersection of two or more of them:

• Algebra. The study of mathematical structures and the operations defined on them — from the solving of equations taught in secondary school through to abstract algebra’s study of groups, rings, and fields, the structures underlying much of modern pure mathematics.
• Analysis. The rigorous study of limits, continuity, derivatives, and integrals — the formal foundation beneath calculus. Real analysis, complex analysis, and functional analysis extend these ideas to progressively more abstract settings.
• Geometry and topology. Geometry studies shape, size, and the properties of space; topology studies the properties of shapes that survive continuous deformation (stretching and bending, but not tearing). Differential geometry, which underlies Einstein’s general relativity, sits at this intersection.
• Number theory. The study of the integers and their properties — primes, divisibility, Diophantine equations. Long considered the purest branch of mathematics, number theory turned out to be foundational to modern public-key cryptography.
• Combinatorics and discrete mathematics. The study of countable, discrete structures — graphs, networks, counting problems, combinatorial optimization — with close ties to theoretical computer science.
• Probability theory. The mathematical study of randomness and uncertainty, and the shared foundation beneath both statistics and much of modern applied mathematics (stochastic processes, mathematical finance).
• Applied and computational mathematics. Mathematical modeling, numerical analysis, and scientific computing applied to problems from physics, biology, engineering, and finance — where an exact closed-form solution often doesn’t exist and the goal is a provably accurate numerical approximation instead.
• Mathematical logic and foundations. The study of formal systems themselves — what can and cannot be proven within a given axiomatic system, set theory, and the foundations underlying all of mathematics. Kurt Gödel’s incompleteness theorems, showing that any sufficiently powerful formal system contains true statements it cannot prove, are the field’s best-known result.

Who Funds Mathematics Research

Mathematics research in the United States is funded primarily through the National Science Foundation, with additional support from mission-specific federal agencies and a small number of private foundations:

• National Science Foundation (NSF). The Division of Mathematical Sciences (DMS), part of NSF’s Directorate for Mathematical and Physical Sciences (MPS), is the primary federal funder of core mathematics research in the US, supporting work across algebra, analysis, geometry, topology, number theory, combinatorics, probability, and applied and computational mathematics, alongside CAREER awards for early-career faculty and postdoctoral research fellowships.
• Department of Energy (DOE). DOE’s Office of Science funds applied and computational mathematics through its Advanced Scientific Computing Research (ASCR) program, which supports the numerical methods, algorithms, and mathematical models needed for large-scale scientific computing on DOE’s high-performance computing systems.
• National Security Agency (NSA). The NSA’s Mathematical Sciences Program funds unclassified, extramural basic research in mathematics — historically including areas such as number theory, algebra, and combinatorics — reflecting the discipline’s long-standing relevance to cryptography.
• National Institutes of Health (NIH). NIH does not maintain a dedicated mathematics institute, but funds mathematical and computational modeling as part of broader biomedical research grants — for example through the National Institute of General Medical Sciences (NIGMS), which explicitly supports mathematical and computational approaches to biological problems.
• Private foundations. The Simons Foundation, through its Mathematics and Physical Sciences division, is one of the largest private funders of mathematics research, supporting Simons Investigators, Simons Fellows, and large multi-institution research collaborations. The Alfred P. Sloan Foundation funds early-career mathematicians through its Sloan Research Fellowships. The Clay Mathematics Institute funds mathematics research and public engagement and administers the Millennium Prize Problems. As with any funding landscape, program scope, budgets, and priorities shift — research offices should confirm current guidelines directly on each funder’s own site before an application, rather than relying on a general overview like this one.

Research Methods and Tools in Mathematics

Mathematical research draws on a mix of traditional proof-based work and, increasingly, computational tools:

• Proof construction. The core method of pure mathematics: starting from axioms and established theorems, constructing a logically valid chain of reasoning that establishes a new result, then subjecting that proof to peer review so other mathematicians can independently verify each step.
• Symbolic computation. Computer algebra systems such as Mathematica, Maple, and the open-source SageMath perform exact symbolic manipulation — simplifying expressions, solving equations, and exploring conjectures computationally before (or instead of) a hand-written proof.
• Numerical and computational methods. Applied and computational mathematicians use tools such as MATLAB, Julia, and Python (with libraries such as NumPy and SciPy) to develop and test numerical algorithms and simulate mathematical models that have no closed-form solution.
• Formal proof verification. Proof assistants such as Lean, Coq, and Isabelle allow a proof to be encoded in a form a computer can mechanically check line-by-line, removing any doubt about a subtle logical gap. These tools are increasingly used to formally verify complex proofs that are otherwise difficult for human reviewers to check exhaustively.
• Preprint and literature infrastructure. The arXiv preprint repository is the dominant venue for rapid, open dissemination of new mathematics research ahead of formal journal publication. MathSciNet (maintained by the American Mathematical Society) and zbMATH are the field’s primary abstracting and reviewing databases for locating prior work.

Career and Training Pathways

Mathematics has a fairly standardized academic training pipeline, alongside strong demand for mathematically trained graduates well beyond academia:

• Undergraduate. A bachelor’s degree in mathematics typically covers calculus, linear algebra, real analysis, and abstract algebra, and serves as preparation either for graduate study or for quantitative roles in industry, finance, or technology.
• Graduate training. A PhD in mathematics typically takes around five to six years: initial coursework and qualifying examinations covering the field’s core areas, followed by original dissertation research conducted under a faculty advisor, usually resulting in a proof of new theorems in a specific subfield. A postdoctoral research position (typically two to three years) is common before a tenure-track academic appointment.
• Professional societies. The American Mathematical Society (AMS), founded in 1888, is the principal professional society for research mathematicians in the US, publishing numerous research journals and maintaining MathSciNet. The Society for Industrial and Applied Mathematics (SIAM), founded in 1951, is the leading society for applied and computational mathematics. The Mathematical Association of America (MAA), founded in 1915, focuses primarily on undergraduate mathematics education and co-administers the William Lowell Putnam Mathematical Competition.
• Career destinations. Mathematics PhDs and graduates work across academia, finance and quantitative trading (as quants and risk analysts), technology companies (as software engineers, data scientists, and machine learning researchers), cryptography and national security (including the NSA), actuarial science, operations research, and mathematics education — reflecting how broadly mathematical training transfers across sectors.

Frequently Asked Questions

What is the difference between pure and applied mathematics?

Pure mathematics studies mathematical structures for their own sake, without requiring an application outside mathematics; applied mathematics develops and uses mathematical tools to solve problems originating in other fields, such as physics, engineering, or biology. Many results discovered as pure mathematics later become essential to applied work, so the boundary is porous rather than fixed.

What is the difference between mathematics and statistics?

Statistics grew out of probability theory, a branch of mathematics, and remains grounded in it, but statistics is widely treated as a distinct discipline because so much of its content — study design, applied data analysis, domain-specific methodology — is not purely mathematical. See CASRAI’s guide to statistics for more detail.

Do you need a PhD to do mathematics research?

For a career as an independent academic research mathematician, a PhD is effectively required. Many quantitative roles in industry — finance, data science, software engineering — make heavy use of mathematical training at the bachelor’s or master’s level without requiring a PhD or involving original mathematics research as such.

What are the Millennium Prize Problems?

The Millennium Prize Problems are seven of the most important open problems in mathematics, named by the Clay Mathematics Institute in 2000, each carrying a $1 million prize for a correct solution. As of this writing, only one — the Poincaré conjecture — has been solved, by Grigori Perelman in 2003, who declined the prize.

How is mathematics related to computer science?

Theoretical computer science — algorithms, computational complexity theory, formal logic, cryptography, discrete mathematics — is essentially a branch of applied mathematics, and the two fields overlap heavily in departments, conferences, and researchers. See CASRAI’s guide to computer science for more detail.

Related CASRAI Resources

This guide is part of CASRAI’s Branches of Science series, which maps the major academic and scientific disciplines along with the research-administration context — funding, methods, and career pathways — that a general encyclopedia entry typically omits. Related discipline guides include statistics, physics, computer science, economics, data science, and, from CASRAI’s current branches-of-science wave, artificial intelligence, astronomy, and industrial engineering.