What Field of Math Should I Study

One press picks an area of mathematics to study, with the kind of question it asks.

What Field of Math Should I Study

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  1. Number theoryWhat is true of the integers, and above all of the prime numbers?

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How it works

Choosing a course, a thesis topic or something to teach yourself? Press the button for one area of mathematics, from number theory to game theory.

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What area of math should I study

Go by the question that pulls you in. Number theory asks about the integers and the primes, topology about what survives stretching and bending, combinatorics about counting, and probability about chance.

What kind of math should I learn first

The usual school sequence in the United States runs pre-algebra, algebra, geometry, a second year of algebra, precalculus, then calculus or statistics. Linear algebra and abstract algebra come after that, for students who major in mathematics and allied fields.

  • The Mathematics Subject Classification files research under 63 top-level classes, so any list of fields is one way to cut the subject among many.

Questions and answers

What type of math should I learn?

Wikipedia’s article on mathematics names these among its areas: number theory (the integers), algebra (operations and the structures they form), geometry (shapes and the spaces that contain them), analysis (approximation and convergence) and set theory (a foundation for all mathematics).

What math should I learn after calculus?

Multivariable calculus extends calculus to functions of several variables, and real analysis develops it rigorously. Linear algebra is central to almost all areas of mathematics.

What is the difference between pure and applied math?

Pure mathematics studies mathematical ideas apart from any use outside mathematics. Applied mathematics brings mathematical methods to fields such as physics, engineering, biology and finance. Results from pure areas may turn out useful later.

What math should I learn for computer science?

Discrete mathematics, which studies integers, graphs and statements in logic. Graph theory, computability theory and computational complexity theory all sit between mathematics and computer science.

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All 74 fields of math

Taught in school 7

  • Arithmetic Numbers and the operations on them: adding, subtracting, multiplying and dividing.
  • Calculus How fast is a quantity changing, and how much of it has built up?
  • Elementary algebra Which values of a variable make a statement true?
  • Euclidean geometry What follows about points, lines and angles from a small set of simple axioms?
  • Precalculus Algebra and trigonometry at the level that prepares a student for calculus.
  • Statistics How should data be collected, analyzed, interpreted and presented?
  • Trigonometry How do the angles of a triangle relate to the lengths of its sides?

Algebra and number theory 11

  • Abstract algebra Sets with operations on their elements: groups, rings, fields and other algebraic structures.
  • Algebraic number theory Questions about the integers and rational numbers, answered with abstract algebra.
  • Analytic number theory Questions about the integers and the primes, answered with methods from analysis.
  • Category theory A general theory of mathematical structures and the relations between them.
  • Commutative algebra Commutative rings, their ideals and the modules over them.
  • Galois theory Which polynomial equations can be solved by radicals? It ties field theory to group theory.
  • Group theory The structures called groups, which recur throughout mathematics.
  • Linear algebra Linear equations and linear maps, and the vector spaces and matrices that represent them.
  • Number theory What is true of the integers, and above all of the prime numbers?
  • Representation theory How can the elements of an abstract algebraic structure be written as matrices?
  • Ring theory Structures with an addition and a multiplication that behave much as they do for the integers.

Geometry and topology 14

  • Algebraic geometry What shapes do the solutions of systems of polynomial equations make?
  • Algebraic topology How can tools from abstract algebra tell topological spaces apart?
  • Analytic geometry Geometry done with a coordinate system.
  • Computational geometry Algorithms that can be stated in terms of geometry.
  • Convex geometry The geometry of convex sets, mainly in Euclidean space.
  • Differential geometry The geometry of smooth shapes and smooth spaces, such as curves and surfaces.
  • Differential topology The coarser properties of smooth manifolds, such as the number of holes in one.
  • Discrete geometry How can points, lines, circles and polygons meet, or be arranged to cover a larger object?
  • Fractals Shapes with detailed structure at every scale, however small, often similar from scale to scale.
  • Knot theory When can one knot with its ends joined be deformed into another?
  • Non-Euclidean geometry What geometry do you get when the parallel postulate is replaced with another?
  • Projective geometry Which geometric properties stay the same under a projective transformation?
  • Riemannian geometry Surfaces and their higher-dimensional kin, with distance measured along curves in the space.
  • Topology Which properties of a shape survive stretching, twisting and bending, with no tearing or gluing?

Calculus and analysis 14

  • Calculus of variations Which function makes a quantity that depends on a whole function largest or smallest?
  • Chaos theory The hidden patterns in systems that are highly sensitive to their starting conditions.
  • Complex analysis Functions whose inputs and values are complex numbers.
  • Differential equations Equations that relate an unknown function to its rates of change.
  • Dynamical systems How complex systems behave over time, usually described by differential or difference equations.
  • Ergodic theory The statistical properties of deterministic dynamical systems, seen through averages over time.
  • Fourier analysis How can a general function be written as a sum of trigonometric functions?
  • Functional analysis Vector spaces that carry a notion of limit, and the linear functions defined on them.
  • Measure theory What do length, area, volume and probability have in common?
  • Multivariable calculus Calculus for functions of several variables in place of one.
  • Numerical analysis Algorithms that give approximate answers to the problems of continuous mathematics.
  • Partial differential equations Equations that involve a function of several variables and its partial derivatives.
  • Real analysis Calculus made rigorous: limits, continuity, differentiation, integration and series.
  • Vector calculus Differentiating and integrating vector fields, mainly in three-dimensional space.

Logic, sets and discrete math 13

  • Coding theory Codes for compressing data and for detecting and correcting errors.
  • Combinatorics Counting, and the properties of finite structures.
  • Computability theory What does it mean for a function to be computable?
  • Computational complexity theory How many resources does a computational problem need, whatever algorithm is used?
  • Cryptography How can a message be kept secure when an adversary is at work?
  • Discrete mathematics Structures that are discrete and not continuous: integers, graphs and statements in logic.
  • Graph theory Points joined by edges, used to model relations between pairs of objects.
  • Mathematical logic What can a formal system of logic express, and what can it prove?
  • Model theory How do formal theories relate to the structures in which their statements hold?
  • Order theory The idea of order: statements such as “this is less than that” or “this precedes that”.
  • Proof theory Proofs treated as formal mathematical objects that can be analyzed in their turn.
  • Ramsey theory How big must a structure be to guarantee that a particular property holds?
  • Set theory Sets, which are collections of objects, and above all the sets that matter to mathematics.

Probability and statistics 7

  • Actuarial science Mathematics and statistics used to assess risk in insurance, pensions and finance.
  • Bayesian statistics Statistics in which a probability expresses a degree of belief in an event.
  • Information theory How can information be measured, stored and communicated?
  • Mathematical statistics Probability theory and other mathematics applied to statistics.
  • Probability theory Probability treated with rigor, built up from a set of axioms.
  • Queueing theory How long will the line be, and how long is the wait?
  • Stochastic processes Families of random variables, used to model things that seem to vary at random over time.

Math put to use 8

  • Control theory How do you drive a dynamical system to a desired state and keep it stable?
  • Game theory Interactions between several agents, modeled as games of strategy between players.
  • Mathematical biology Mathematical models of the structure, development and behavior of living systems.
  • Mathematical economics Mathematical methods used to state theories and analyze problems in economics.
  • Mathematical finance Mathematical models in finance: pricing derivatives, and managing risk and portfolios.
  • Mathematical optimization Which is the best choice, by some criterion, from a set of alternatives?
  • Mathematical physics Mathematical methods developed for use in physics.
  • Operations research Analytical methods for better management and decision-making.