71 Types and Applications of Math

So you may be wondering what’s the point of all this high-level mathematics you might be learning. Or maybe you are a math major trying to figure out where math could take you. Or maybe you are just curious about what kind of math is out there. Here I am going to answer these questions in a short and concise list, so here we go!

1. Calculus
   • Probably the most common higher level math people talk about. Calculus is where you will learn all about derivatives and integrals. Derivatives aim to provide information regarding instantaneous rates of change, while integrals give summation estimations, like the area under a curve for example.
2. Multivariable Calculus
   • As is suggested from the title, Multivariable Calculus simply expands Calculus concepts to multiple dimensions, though mostly just three dimensions to represent the real world mathematically.
3. Linear Algebra
   • Another very commonly used area of math, Linear Algebra involves the study of matrices and vectors. Here you can expect to learn about things like the dot product, cross product, Kronecker product, and more! This material is used in many different fields and will be mentioned later on in this list.
4. Set Theory
   • Set Theory studies what a collection of objects is, in a more formal manner. This material provides the basics necessary to understand a lot of the math mentioned in this list.
5. Group Theory
   • The beginning of abstract algebra, in Group Theory you study groups, which is a collection of objects (numbers, variables, etc.) which have a few specific properties. The most notable property would be an operation, such as addition or multiplication. Theorems and lemmas are then derived from these definitions and proved extensively.
6. Ring Theory
   • Ring Theory is an extension of group theory. You will see that often the operation for a group is addition, but for rings you will have both addition and multiplication. Rings have their own theorems and lemmas, some of which are natural extensions from group theory, but most are not.
7. Field Theory
   • Field Theory is an extension of ring theory, thus making it another extension of group theory. A field is simply a set of objects with even more properties than a ring. Some common examples would be the set of real numbers, or the set of complex numbers.
8. Galois Theory
   • Galois Theory explores the study of the roots of polynomials – a concept you may first learn in calculus. It is in fact a blend of group theory and field theory.
9. Graph Theory
   • Graph Theory studies the notion of a graph, which is a collection of nodes with edges (lines) attaching nodes together. Graphs can be used to model many problems, though most notably the traveling salesman problem, which seeks to optimize the distance traveled between many cities.

10. Category Theory
    • Category Theory is abstract math at its finest. Here you generalize the study of math by studying categories. Categories are composed of nodes, which are called objects, and have arrows pointing between different nodes. You can think of it as a generalized blend of abstract algebra and graph theory.
11. Complex Variables
    • Complex Variables deals with imaginary numbers, which is just the square root of negative numbers. This is especially important in physics, but shows up in all sorts of different fields.
12. Real Analysis
    • Real Analysis studies things like sequences and series. It explores properties such as limits, continuity, and more. Calculus concepts also come from this subject matter in a more general sense.
13. Complex Analysis
    • By bringing complex variables into real analysis, you will get Complex Analysis. It is essentially a more abstract study of the typical complex variable material you may come across.
14. Functional Analysis
    • A subject of analysis, it focuses more on the study of vector spaces, and linear functionals and operators. As a fairly advanced subject, expect to see material from this subject matter also in subjects like physics, although in a more simplified and less abstract manner.
15. Measure Theory
    • Measure Theory generalizes what you may already know of as measurements, such as length or area. An example for further consideration would be the Lebesgue Measure.
16. Combinatorics
    • Combinatorics is the study of counting. While that may make it sound simple, it is in fact still a challenging subject, and crosses paths with other subjects like graph theory.
17. Number Theory
    • A popular subject, and one of the more fun ones, Number Theory is essentially the study of integers. Here you will study things like modular arithmetic, prime numbers, and also rational numbers in detail.
18. Algebraic Number Theory
    • Algebraic Number Theory expands number theory by using concepts from ring or field theory. A well known fact is that polynomials of degree 5 and higher can’t be factored, and it is here where this fact will be proved.
19. Cryptography
    • Cryptography uses math like number theory to protect data behind encryption. Here you will learn about ciphertext, and how to encode and decode it.
20. Ordinary Differential Equations (ODE’s)
    • Differential equations are just what it sounds like, equations with derivatives involved. ODE’s can be seen as the next step up from calculus, but essentially here you will learn how to solve such equations.
21. Partial Differential Equations (PDE’s)
    • PDE’s are just ODE’s but with partial derivatives in the equations also. This is a heavily researched subject as many partial differential equations govern certain physical situations, but with no real solution.
22. Geometry
    • In simplistic terms you can think of Geometry as the study of shapes, but more generally it studies the properties of lines, surfaces, and similar things in higher dimensions.
23. Topology
    • In Topology, you will look at certain objects and perform certain transformations on them. Through these transformations certain properties of the object will be preserved, and these properties are primarily what will be studied.

24. Mathematical Physics
    • Mathematical Physics is typically just a subject with mathematical material relevant only to a physicist. You will see a wide variety of subjects here, but usually not explained in great detail.
25. Kinematics
    • Moving on to more physics-related subjects, Kinematics is usually an introductory subject in the field of physics, often taught first in high school. It is simply the study of objects and how forces applied to them affect them. Basic calculus is typically the only math you will need here.
26. Special Relativity
    • Special Relativity is typically the first subject in physics that you will come across that truly challenges the way you think about the real world. It is the study of how high speeds affect objects, and even time itself! It is here you will be introduced the infamous equation E=mc². Not much higher level math will be used here, though calculus will still likely be needed.
27. Electrostatics
    • In electrostatics you will study electric charges but with no movement. The math involved here will primarily be multivariable calculus, though some linear algebra may also be needed.
28. Thermodynamics
    • To study Thermodynamics, which is essentially the study of temperature and heat, you will find multivariable calculus and basic linear algebra to be useful, although not as much as other physics subjects.
29. Condensed Matter Physics
    • There are three states that a substance may be in, namely gas, liquid, and solid. Condensed Matter Physics is the study of when a substance is in its solid state. Like most physics subjects, calculus and linear algebra will be seen here, although things like group theory and topology also have applications in this field.
30. Solid State Physics
    • Solid State Physics is a subset of condensed matter physics, and is the largest subset in the field. As such, similar mathematics will be seen here, although other physics subjects like electrodynamics will be used as well.
31. Electrodynamics
    • Speaking of Electrodynamics, this subject is an extension of electrostatics, but now we have moving charges, hence the name. Expect more rigorous calculus in this field.
32. Statistical Mechanics
    • Statistical Mechanics applies probability and statistics to model the behavior of large numbers of atoms or molecules. Concepts from thermodynamics will also be useful here, and hence so will the math needed for it.
33. Classical Mechanics
    • Physics can be thought of as having two different major branches. One of which is Classical Mechanics, which studies the behavior and motion of large objects in a generalized manner. The math used here is primarily calculus and linear algebra, as well as differential equations.

34. Quantum Mechanics
    • The other half of physics is Quantum Mechanics, which deals with objects on a much smaller scale. The math used here will be standard calculus as usual for physics, but also more advanced linear algebra, as well as probability and complex variables.
35. Quantum Computing
    • A specific application of quantum mechanics, Quantum Computing uses the principles of quantum mechanics to make computers run on qubits instead of bits. Of course all the math seen in quantum mechanics will be used here, though depending on which route you take in this field, things like group theory, number theory, or even graph theory will be needed.
36. Acoustics
    • Acoustics studies mechanical waves in all three different substance states, essentially meaning it studies sound. The math seen here may include calculus as well as differential equations.
37. Optics
    • Simply put Optics is the study of light and its properties. It is here where you will learn about things like lasers and such, thus requiring the basics again like calculus and linear algebra as well as geometry and differential equations.
38. Quantum Optics
    • A more advanced version of optics, it simply applies the concepts in quantum physics to optics to understand what is called photons. You can expect the math here to be the same as in both quantum physics and optics.
39. Particle Physics
    • Particle Physics studies elementary particles and the interactions they have with each other. Various mathematics can be seen in this subject, such as calculus, group theory, and Geometry.
40. Nuclear Physics
    • The study of an atom’s protons and neutrons and their interactions. Being a more advanced subject, the math needed here will vary from group theory to calculus to complex variables.
41. Atomic Physics
    • Here you study the atom as a whole and not just its individual parts like in nuclear physics. Though the math needed here will still be similar to that of nuclear physics.
42. Medical Physics
    • Medical Physics involves the use of x-rays, ultrasound, and more to perform radiation therapy. Basic math and physics like calculus and linear algebra will be the bulk of the background material seen here.
43. General Relativity
    • Essentially the study of gravity, General Relativity is a more advanced topic and so may require mathematics involving not only calculus and linear algebra, but also differential equations and analysis.
44. Astronomy
    • The study of space basically, Astronomy is a popular subject in physics, but rather advanced and thus can be seen to use mathematics like calculus, linear algebra, geometry, differential equations and more.
45. Chemistry
    • Chemistry as you may already know is the study of matter, in particular dealing with interactions of the known elements. Most math seen in chemistry is fairly basic, and learning calculus for nothing more than mathematical maturity may be enough.
46. Quantum Chemistry
    • Simply applying the principles of quantum mechanics to chemical systems, you get the study of Quantum Chemistry. Naturally the math needed here will mostly be the same as quantum mechanics.
47. Electronics
    • A subject in physics but also in engineering and robotics, you will see that learning about electrical currents requires mostly calculus, but also knowledge from complex variables and electrodynamics.
48. Statistics
    • While Statistics is in a way a different kind of math, it is worth mentioning briefly here in a broad sense. It is simply the analysis of large amounts of data, with the intention of drawing some kind of information of conclusion from the data.
49. Probability
    • A subset of statistics, it is the study of the likelihood of some event occurring given certain or no conditions.

50. Programming
    • In a broad sense programming itself doesn’t necessarily require a lot of math, and there are a ton of programming applications and jobs that do not use much math, however the logical thinking when programming is similar to that of mathematics subjects.
51. Supervised Learning
    • Now we venture into the world of machine learning, by first mentioning perhaps the simplest subset, Supervised Learning. In this subject the goal is to make a predictive algorithm given what is called “labeled data”. Although some calculus and linear algebra is used here, it can be glossed over, making statistics and probability the primary mathematics needed.
52. Unsupervised Learning
    • Similar to supervised learning, Unsupervised Learning also makes use of the same mathematics, just this time with “unlabeled data”.
53. Neural Networks
    • Neural Networks is another part of machine learning, and perhaps the first subject in this field where linear algebra and calculus will be more prevalent. This subfield also aims to also make predictive algorithms but this time to capture “non-linearity” aspects of a model.
54. Deep Learning
    • Deep Learning basically involves applying more intensive Neural Networks, and thus the calculus and linear algebra backgrounds become even more important.
55. Computer Vision
    • Computer Vision can be thought of as a special kind of deep learning, where we apply the concepts and algorithms to images to make predictions on them. As such, similar mathematics can be seen here.
56. Natural Language Processing (NLP)
    • Another type of application of deep learning, NLP makes algorithms regarding language, like reading and detecting spam in emails, or something else like voice recognition. Again, calculus and linear algebra will be important here.
57. Reinforcement Learning
    • Reinforcement Learning is a type of machine learning where an algorithm will learn as it continues to collect data. Calculus, linear algebra, and statistics will all be needed to grasp these concepts, but depending on the type of problem other higher levels of math may be useful.
58. Artificial Intelligence
    • Surely Artificial Intelligence needs no introduction, though briefly it is just the goal of creating some kind of computer that can perform tasks that normally only humans could do. This is a broad field, so the math needed for this subject will vary depending on what you choose to specialize in.

59. Robotics
    • Though mostly similar to artificial intelligence, a lot of robotics problems will involve you building the physical robot which will require electrical circuitry knowledge, and hence higher levels of math like complex variables and calculus.
60. Self-Driving Cars
    • A specific type of robotics, specializing in Self-Driving Cars will demand from you mathematics and physics like multivariable calculus, linear algebra, classical mechanics and more.
61. Data Mining
    • In Data Mining you will strive to search large data sets to find new relevant information. Similar to machine learning subfields, more basic calculus, linear algebra, statistics and probability will be seen in this field.
62. Game Design
    • Game Design is (obviously) where to design video games. Oftentimes this means you will be modeling the real world, thus requiring all the basic mathematics and physics such as calculus, linear algebra, kinematics, probability, and so on. Sometimes higher levels of math may be needed to model more complex physical systems.
63. Mechanical Engineering
    • A mechanical engineer’s job involves designing and maintaining mechanical systems, and this can involve a wide variety of mathematics ranging from multivariable calculus to differential equations. Things like complex variables and linear algebra will also be needed.
64. Electrical Engineering
    • Electrical engineering is just a more focused and applied field of electronics, and such the mathematics and physics needed will largely be the same (that is multivariable calculus, electrodynamics, etc.).
65. Aerospace Engineering
    • In Aerospace Engineering, as the name implies, the focus is on building machines with flying capabilities. The math needed for this type of engineering is the same as the others mentioned.
66. Economics
    • Economics is a social science that analyzes market data regarding the production and distribution of goods and services. The math here will often be basic statistics, probability, but also linear algebra and calculus. Sometimes even higher level math may be seen, though not often.
67. Finance
    • Finance is basically the study of money, and the math needed will be similar to the math seen in the study of economics.Note that it is common to see some machine learning concepts be used in this field as well.
68. Actuarial Science
    • In Actuarial Science, you will analyze risk and uncertainty in business settings. The math used here is also similar to that of economics.
69. Philosophy
    • Similar to programming, Philosophy also doesn’t necessarily involve much math, if at all, however the logical thinking is again very similar to that of mathematics subjects.
70. Music
    • This may be an unexpected addition to this list, but there is some study of applying graph theoretic concepts to find relations between different chords and such. Naturally, graph theory is the main mathematics used here.
71. Teaching
    • An obvious application of mathematics, and mentioned only for completionism’s sake, teaching math is of course necessary for others like you and me to learn about it.