Mathematics and applied mathematics are a wide range of subjects, and the knowledge that needs to be studied in depth includes the following aspects:

1. Advanced Mathematics: This is the basis of mathematics and applied mathematics, including calculus, linear algebra, probability theory and mathematical statistics. This knowledge is the basis of studying other branches of mathematics in the future, and it is also an important tool to solve practical problems.

2. Discrete mathematics: Discrete mathematics is a branch of mathematics that studies discrete structures and their properties, including graph theory, combinatorial mathematics and logic. This knowledge is widely used in computer science, information theory and other fields.

3. Real variable function and functional analysis: This is an important branch of modern mathematics, which mainly studies functions and their properties on real numbers or complex numbers. This knowledge has important applications in the fields of partial differential equations and numerical analysis.

4. Ordinary differential equations and dynamic systems: This is a subject that studies the properties and dynamic behavior of solutions of ordinary differential equations. This knowledge is widely used in physics, engineering and other fields.

5. Numerical analysis: This is a subject that uses numerical methods to study approximate solutions of mathematical problems, including interpolation, approximation, numerical integration, numerical differentiation and so on. These knowledge have important applications in scientific calculation, engineering simulation and other fields.

6. Operational Research and Optimization Method: This is a mathematical theory and method to study decision-making problems, including linear programming, nonlinear programming and dynamic programming. This knowledge is widely used in economic management, engineering technology and other fields.

7. Probability theory and mathematical statistics: this is a discipline that studies the regularity of random phenomena, including the basic concepts of probability theory, random variables and their distribution, the law of large numbers, the central limit theorem and so on. This knowledge has important applications in insurance, finance, biomedicine and other fields.