Math Topics (A Condensed Guide to Mathematics)

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Math Topics back view of man standing in dark grey room whose walls are painted different formulas

Table of Contents

No matter who you are, you’ve probably had some math classes in your life. After all, math is an important subject not only for success in technical subject areas, but also for giving your mind a workout and making it stronger. You might be taking such a class right now and need some help with that workout. Or you might be about to choose a class but can’t decide which direction to go. Or maybe you’re just interested in the subject and want to see what all the field can offer you. Well, spoiler alert—it’s a lot. The number of math topics in the world just keeps getting bigger and bigger through time, especially now with the importance of math for many technical professions. But this article condenses the list a bit to help you get a better look at the field.

Foundation Math Topics

The following topics don’t focus so much on foundations of math methods or calculations. Rather, they encompass all of the theories that hold up math as a way of thinking. These topics explore the logic of the field of math as it is used to describe the world around us. They analyze how we use that logic and why. In these topics, you’ll find answers to questions about what drives math and how to properly conduct mathematical tasks in order to reach the goals of the field. And learning about these topics can make it easier to understand the math problems we all learn in school by making the purpose and meaning behind the problems clear.

Category Theory

Category theory is a mathematical theory of structures and of systems of structures. It provides a conceptual framework that allows us to see the universal components of a family of structures of a given kind and the relationships between those structures.

Information Theory

Information theory is the study of the storage and transmittal of information. It uses techniques from mathematics, statistics, and physics and has applications in computer science, cryptography, and neurobiology, as well as psychology and linguistics.

Mathematical Logic

Mathematical logic is a field of study that unites the studies of formal logic with mathematics. Problems in mathematical logic have been motivated by the development of axiomatic systems for arithmetic and geometry. Mathematical logic is also connected to theoretical computer science.

Philosophy of Mathematics and Related Topics

The topics of math and philosophy aren’t as far apart as some might think. The philosophy of mathematics looks at the foundations of math itself by analyzing assumptions it makes and the implications of its methods and results. This can help us determine the role of math in peoples’ lives in areas such as its influence on science, technology, and logic itself.

Math Topics in Set Theory

One of the main topics in the philosophy of math in the 20th century is set theory. It looks at the relationship between math and logic in the context of collections of objects or numbers (called sets). The topic is often considered a foundation for mathematics due to its ability to define nearly all objects in the field.


Algebra Math Topics

Algebra is a collection of math topics in which the main goal is to determine an unknown value or values. It uses various signs and symbols to represent known and unknown values in equations and inequalities. (Think of statements that use the symbols =, >, <, ≠). Manipulations of these equations can help us to solve for the unknown values. But, more importantly, algebraic rules also work in real life! In sciences and technical fields, the numbers and symbols of equations represent real things. Fields that commonly use algebra include engineering, computer science, physics, accounting, construction, and business.

Group Theory

Group theory is a branch of abstract algebra that studies algebraic constructs called groups. A group is a set of elements together with a binary operation such as addition or multiplication. Group theory has applications in physics and chemistry as well as in cryptography.

Commutative Algebra

Commutative algebra is a field of mathematics that studies algebraic structures called commutative rings. An example of a commutative ring is the set of integers. Commutative algebra has applications to algebraic number theory and algebraic geometry.

Operator Algebras

The study of operator algebras is a field of functional analysis that studies continuous linear operators on a topological vector space. An operator algebra is an example of a non-commutative ring. Operator algebras have applications in differential geometry and quantum field theory.

Representation Theory

Representation theory is a field of mathematics that represents algebraic objects as matrices and the algebraic operations as matrix addition and multiplication. It reduces abstract algebra problems to more simple linear algebra problems, thus making calculations more doable.

Rings and Algebras

This math topic also circles back around to sets. If a set meets certain characteristics, then it is called a ring. There are various types of rings, each with special characteristics and in terms of its makeup. And, in this case, and “algebra” refers one of those rings.

Universal Algebra

Also called general algebra, this field looks at algebraic structures. These structures, also called “algebras,” are sets on which a collection of operations is performed. This is unique because other areas of math do not so much study these structures themselves. They often look at models of structures in which examine particular examples within the structure rather than the whole structure.


Analysis Math Topics

Mathematical analysis is a branch of topics that comes from calculus. It deals with limits and theories relating to them, usually in the context of complex and real numbers and functions. Put another way, the analysis studies functions and their generalizations using the method of limits. This means it is first important to understand functions and limits so that you can relate them within analysis. Other important concepts in the field are those of metric spaces and sequences. Using these concepts, you can explore the branches of real, complex, numerical,, and functional analysis, as well as differential equations and measure theory.

Calculus: A Commonplace Math Topic

This classic math topic is the study of continuous change. The most important branches include differential and integral calculus. Differential calculus looks at rates of change represented by slopes of curves when data is put into a graph. Integral calculus then looks at the areas under and between the curves of data, which shows accumulation. This is a common class that students take as early as high school which introduces them to concepts that become the foundations of higher-level math.

Classical Analysis

The analysis of ordinary differential equations applies analytical methods to solve rate-of-change equations involving one variable. The analysis of ordinary differential equations is motivated by problems in physics and also has applications to various areas of engineering.

Differential Equations

Differential equations is a topic many students encounter in university classes. It relates functions to their derivatives, which is useful for relating physical quantities in real life with their rates of change. Thus, these relationships are very common and very useful in fields like physics, engineering, and economics.

Complex Variables

Complex analysis is a field of mathematics that studies functions that act on complex numbers. It also has applications to many areas of mathematics and physics, including number theory and thermodynamics. Additionally, complex analysis also has applications in certain engineering fields.

Partial Differential Equations: Common but Advanced Math Topic

The analysis of partial differential equations applies analytical methods to solve rate-of-change equations involving multiple dimensions. The analysis of partial differential equations is motivated by problems in physics and differential geometry, and has applications to various areas of engineering.

Functional Analysis

Functional analysis is a field of mathematics that studies functions of functions; in other words, operators whose inputs are themselves functions. Functional analysis seeks to classify functions into broad classes, such as those that can be differentiated or those that can be integrated.

Harmonic Analysis

This math topic looks at how functions and signals are represented as the superposition of waves. It does this by studying and generalizing things such as Fourier series and transforms. It is an important topic in areas such as number theory, signal processing, neuroscience, and quantum mechanics.


Discrete Math Topics

In math there are continuous mathematical structures and then those which are not continuous, aka discrete. Rather than varying smoothly like real numbers, objects in discrete math, including integers, logic statements, and graphs, have distinct and separated values. Other areas such as calculus and Euclidean geometry deal with continuous math, and therefore are not included in discrete math. In addition to a lack of continuity, discrete math deals with countable sets. These sets can be finite or infinite, leading to a number of focuses and applications. One of these applications is digital computers. They operate in discrete steps while also storing data in discrete bits.

Combinatorics Math Topics

Combinatorics is a field of mathematics primarily concerned with counting. In addition to the obvious statistical applications, combinatorics is used frequently in computer science to analyze the efficiency of algorithms. Graph theory is also a subspecialty of combinatorics.

Graph Theory Math Topics

This theory is a mathematical field that models systems of interrelated objects. The graph consists of the objects, nodes, and the relations between them, edges. Graph theory has many applications in discrete mathematics, as well as other fields such as neuroscience.

Game Theory: A Popularized Math Topic

Game theory is the study of decision-making models between rational, independent, and competing parties in a strategic setting. It has applications in economics and political science as well as information technology and evolutionary biology. You may have heard of this topic from some popular movies such as A Beautiful Mind.

Order Theory

This is a fairly simple math topic, but also one that is very important in many parts of the field. It looks at and describes the order of statements using what are called binary relations. This basically means it compares two sets by using terms such as “greater than” and “equal to.”


Geometry Math Topics

Geometry is a branch of math that relates to many topics, from the abstract to the realistic and practical. It looks and shape and size as well as the relative position of objects and figures in space. This can help us understand how the natural world works, and is critical in creating man-made structures. But it also delves into spaces that are beyond the natural world that we can see. Certain types of geometry look at other dimensions and strange properties that are hard to even wrap your mind around.

Algebraic Geometry

One of the above-mentioned abstract topics in this area of math is algebraic geometry. It looks at abstract algebraic techniques that come from commutative algebra, and the goal of these techniques is to solve problems about sets of zeros of multivariate polynomials. It’s a central topic in modern mathematics that has an important role in fields such as topology, complex analysis, and number theory.

Metric Geometry

Metric geometry is an approach to geometry based on the concept of measurement in a topological space, where the distance between all elements of the set is defined and gives rise to a topology. This topic also has applications to Riemannian geometry and group theory.

Differential Geometry

Differential geometry is a field of mathematics that applies methodology from differential and integral calculus and linear algebra to study problems in geometry. Due to the importance of geometry in physical sciences, it has applications in physics, chemistry and various fields of engineering.

Symplectic Geometry

Symplectic geometry is an even dimensional geometry. It measures the sizes of 2-dimensional objects rather than the 1-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry. Symplectic geometry is naturally associated with the field of complex rather than real numbers.

Euclidean Geometry

This math topic has been around since Ancient Greece where it was founded. Similarly to philosophy, the method consists of deducing theorems from original small intuitive axioms. This means that the geometric principles are fueled by logic and intuition, unlike analytic geometry which, rather, uses coordinates and algebraic formulas.

Finite Geometry

This is a type of geometry in which there are only a finite number of points. This is in contrast to Euclidean geometry, in which a line has an infinite number of points. For instance, and example of finite geometry would be the graphics on computer screens. In this case, the finite points are the pixels which make up the graphics.


Number Theory Math Topics

Number theory is a field of mathematics that seeks to uncover properties of and relationships among positive integers. It has applications in computer science and encryption theory. Many number theory problems are easily understood by anyone, although the solutions may require sophisticated mathematics.

Arithmetic

Arithmetic, also sometimes called number theory, is one of the first math topics that everyone learns in school, and one of the few that even non-mathematicians carry with them throughout life. It is the study of numbers, their properties, and how they behave when we manipulate them. The manipulation in this case is just operations like adding, subtracting, multiplying, and dividing.

Algebraic Number Theory

This math topic is a branch within number theory that uses abstract algebra to study numbers and their generalizations. The branch uses methods of expression such as algebraic number fields, rings of integers, and function and finite fields.

Analytic Geometry

Also known as Cartesian or coordinate geometry, this topic is the study of geometry in coordinate systems. These systems can be in a number of different dimensions. Thus, through analytic geometry, we can define and represent shapes in the multidimensional space such that we can extract information from them.

Diophantine Geometry

This is a sub-branch of arithmetic geometry in which algebraic points are studied. These points have coordinates of integers, rational number, and the generalization of them. It is an important branch of study if you want to work on something such as the theory of Diophantine equations. To learn more about the general approach, check out material on Faltings’s theorem.


Topology Math Topics

Math topics in topology look at what happens when objects are deformed. While tearing is not counted as a deformation, other deformities can include twisting and stretching. Certain objects can be related to one another based on the possible deformations of their shape. For example, a circle can be stretched into an ellipse, therefore the two shapes are “topologically equivalent.” This leads to some interesting ideas such as a circle being able to be treated as an object in its own right no matter how it is situated in space. Thus, this means the field of topology has important implications for other topics such as space-time, fractals, knots, manifolds, and more.

General Topology

General topology, or point-set topology, is the foundation of all other branches of topology. It is the field of mathematics that classifies the subsets of a set into those that are open, and then studies concepts such as continuous, connected, or compact in terms of the open sets.

Algebraic Topology

Homology is a field of mathematics that analyzes and classifies algebraic objects such as groups in terms of other algebraic objects such as topological spaces. K-Theory is a specific application of homology on certain topological spaces, i.e. locally compact Hausdorff spaces.

Differential Topology

As topology deals with shapes and objects, differential topology specifically looks at smooth manifolds which are differentiable. It is closely related to differential geometry, but they look at different notions of manifolds. Differential geometry deals with metrical notions of them, whereas differential topology deals with the nonmetrical notions.

Geometric Topology

Geometric topology starts getting a bit more complicated as it combines objects and adds dimensions. Like its differential counterpart above, it also looks at manifolds; however, rather than differentials, it looks at what happens when manifolds are combined or “embedded” into one another.


Applied Math Topics

Topics in the applied mathematics are often less abstract and more practical for everyday needs. Popular and very important fields such as science, business, engineering, and computers all require applied mathematics. In fact, most fields can benefit in some way or another. For example, the social sciences often rely on statistics. Thus, the study of the applied math specifically aims to make advancements in these fields which apply to solving current problems. But these practical developments can have impacts in other areas as well. Additionally, many math theories and pure math have been influenced by advancements in applied math, and vice versa.

Astrostatistics

Astrostatistics applies data mining and statistical analysis to the vast amount of information gathered by automated scanning of the cosmos to link astronomical data with astrophysical theory. It is involved many branches of statistics such as nonparametrics and multivariate regression.

Biostatistics

Biostatistics applies statistical analysis to a wide range of topics in biology. It encompasses both the design of biological experiments and the analysis and interpretation of data from those experiments. However, medical biostatistics, a subspecialty of biostatistics, is exclusively concerned with medicine and health.

Dynamical Systems

This is a field of mathematics that studies systems of functions that relate position to time. Dynamical systems studies the properties and solutions of the motion equations of systems and has applications to physics, biology, and chaos theory.

Probability

Probability talks about the likelihood of something happening. It is related to statistics in that it often uses statistical methods such as density functions. To put it simply, probability puts events on a scale between 0 and 1 where 0 means the event is impossible while 1 means it is certain to happen.

Statistics

Statistics is a science that deals with analysis, classification, interpretation, and collection of numerical facts or data which are, by use of mathematical theories of probability, regularity of aggregates, and imposes or less disparate elements. It can also be defined as a branch of mathematics that deals with the organization, analysis, data collection, presentation, and interpretation of data.

Spectral Theory

Spectral theory is a field of functional analysis which studies the structure of a linear function on the basis of its spectral properties such as the location of the spectrum, the behavior of the resolvent and the asymptotics of its eigenvalues. It extends the concepts of properties associated with a square matrix to

Operations Research

Operations research applies mathematical models to the management of systems in business, industry, government and other enterprises. Applications of operations research include supply chain management and other areas where the efficient allocation of scarce resources is important.

Quantum Algebra

Quantum algebra is a field of mathematics that generalizes commutative algebra, especially commutative algebras arising from Lie theory, to the non-commutative case. The study of quantum algebra is motivated by, and has applications in, quantum physics.

Mathematical Physics

Physics uses a lot of math, and some of that math is pretty specific to just physics. This type of math is called mathematical physics, and helps develop methods that are ideal for solving problems in physics. Important topics in this area of math include classical mechanics, partial differential equations, and statistical mechanics.


Computational Math Topics

In these math topics, the word “computational” can mean two things. It can either refer to making computations in the field of applied math, or it can refer to the use of computers in math. In the second sense, “computers” refers to assistive technology which uses programs and algorithms to solve complex problems. But algorithms and computations in general are important in both senses of the term, and form the main focus of the field. Through these aspects of computational math, many areas benefit. For example, almost every science and many other technical fields benefit from advancements in computing abilities.  

Cryptography

Cryptography is a mathematical field that uses computational algorithms to encrypt information. It is used extensively in modern daily life as a way to keep data secure, protect its integrity, and authenticate originators of transactions. It is one of the more advanced topics in information security.

Computer Science

We tend to think of computer science as the study of digital computers, but in reality it deals with general computation; it just so happens that digital computers are the biggest part of general computation today. The field looks at data processes and the representation of said data via programs.

Optimization

Also called mathematical programming, this topic uses math to select the “best” option from a set of available options. This often means finding the extremes in a range of values. That is, you may want the most money or the least amount of work. However, there are a number of more complex situations and methods.

Computer Algebra

As discussed earlier in this article, algebra uses variables and symbols to solve for unknown quantities. Adding “computer” to the mix just means that algorithms and software are used to explore more advanced math expressions. The use of the computers allows for routine operations and the simplification of complex expressions.

Numerical Analysis

Numerical analysis is a mathematical field that develops, analyzes, and implements approximation algorithms to determine numerical solutions to real-world mathematical problems while quantifying errors. It has applications in all fields of engineering and the physical sciences.


Other Math Topics

It really seems like the list of math topics goes on forever. But that makes sense when you think about how math is basically the basis of everything we know in the universe! It surrounds all the time, and we use it every day in some form or another. Even if it’s just calculating taxes or judging the angles and geometry of roads while we drive. Apart from actively studying the technical aspects of math, other topics explore a different side of the field. The history of math is long and rich, and continues today as we try to understand the best ways to help new generations learn it. There are also some fun sides of math—you just have to look for them!

History of Mathematics

The history of mathematics explores the development and research of mathematical topics, what questions gave rise to those topics, and the mathematicians whose research advanced the field. The history of mathematics also describes the relationships between mathematical theory and other sciences.

Recreational Topics in Math

Math doesn’t always need to be put to use in science or physics or other fields. Some people just like to do math for fun! And this type of math doesn’t just include high-level areas. It also includes puzzles and games that pretty much anyone can play. You can check out some of the fun things to do in math at the Mathematical Association of America website.

Math Topics in Education

Because of the universal applications of math, it generally forms a central part of all formal schooling. From elementary school to college (and beyond), math classes are just part of life. But many people find the subject a bit difficult, so studying the best ways to help students understand math and improve their skills is an entire area of study in itself.

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