There are some differences in each school of mathematics department, but there are several basic core courses that need to be studied seriously: topology, algebra and analysis are the three cornerstones, and it will be annoying if you don't learn well. The history of mathematics is an easily overlooked part. In fact, knowing more will be very helpful for learning mathematics.

Analysis direction: from mathematical analysis to real analysis and complex analysis, to functional analysis, and then to the establishment of probability theory based on functional. These are also the theoretical basis of many commonly used mathematical tools such as transformation analysis.

Geometric direction: you can learn differential fashion directly. If you study Riemannian geometry, you can look at general relativity. This will cover tensor analysis or something.

Other commonly used or more important: differential equation, Lie algebra, game theory, number theory, mathematical logic, graph theory, finite element analysis.

The department of mathematics studies physics, and perhaps it should focus more on mathematical and physical methods than physical concepts like the department of physics.

Learning computer, C language is generally necessary. The so-called discrete mathematics mainly covers mathematical logic, graph theory and number theory in mathematics, which are not as good as learning alone.

Advanced mathematics, linear algebra and complex variable functions are basically non-mathematics majors, and analytic geometry usually helps to lay the foundation in the first year.