Higher mathematics is relative to elementary mathematics, the main difference is whether to introduce the concept of change or movement. In other words, static mathematics is elementary mathematics and dynamic mathematics is advanced mathematics, so it is usually advanced mathematics from the perspective of calculus, and it is better to speak from the concept of function. But now the term advanced mathematics refers to the name of a mathematics course for non-mathematics majors in universities, which usually involves calculus, linear algebra and spatial analytic geometry.
Discrete mathematics is relative to continuous mathematics, mainly based on whether the research object is continuous or not. From this perspective, ordinary calculus is continuous mathematics. But the word discrete mathematics, like advanced mathematics, is now more used to refer to the name of a mathematics course for non-mathematics majors in universities. Its content mainly involves number theory, graph theory, optimization, group theory and other issues, and is usually a compulsory course for computer majors.
Random mathematics is relative to non-random mathematics, mainly based on whether the research object is random or not. Randomness is a kind of uncertainty, so there is a broader classification called deterministic mathematics and uncertain mathematics, which also includes an uncertainty called fuzziness. Anything involving randomness can be classified as stochastic mathematics, such as probability theory, stochastic process, stochastic differential equation, etc. Others, such as calculus and linear algebra, are considered non-random mathematics.
Regarding the connection, I personally think that as long as there is a connection between the branches of mathematics, many methods and skills are interlinked. For example, series can be regarded as discrete form of integral, and then many problems of * * * can be analogized. According to whether the sample space is countable or uncountable, the calculation techniques of discrete form and continuous form can be used respectively.
In terms of specific disciplines, I think the current classification on Wikipedia is more appropriate. That is, according to whether the research object is a problem in the field of mathematics itself or in other disciplines, it can be divided into pure mathematics and applied mathematics. Each category can be subdivided into many subcategories. There are many subclasses. There are two subcategories of applied mathematics, namely statistical mathematics and computational mathematics. There are many subcategories under pure mathematics, but there are at least algebra, geometry and topology, analysis, set theory and so on in the big category.
Incidentally, from the perspective of mathematics learning, the specific content of mathematics is only one aspect of mathematics learning, and the maturity of mathematics is also very important. For example, topology, many of its contents don't need you to have much advanced mathematical knowledge to prepare, but if you don't have a certain degree of mathematical maturity, you will often find it difficult to think. In addition, the three traditional branches of mathematics, algebra, geometry and analysis, are more or less different in thinking methods. Some people are better at algebra, while others are better at analysis. When they encounter specific math problems, they can think from several angles.
Source: Zhihu netizen