For analysis, the courses include: mathematical analysis (the most basic), complex variable function, real variable function, functional analysis and so on. As you said, advanced mathematics is a simple version of mathematical analysis.
In algebra, the courses are: advanced algebra (the most basic), modern algebra (also called abstract algebra) and so on. Higher algebra includes linear algebra and polynomial algebra. Linear algebra (for example, f(x)=Ax+b is called linear because it is a straight line) studies straight lines.
Using polynomial (including not only linear function, quadratic function, but also higher-order function) to replace a very complex function, the result is also very satisfactory.
For geometry, it is mainly analytic geometry.
Randomness, including: probability theory, mathematical statistics, stochastic processes, etc.
Other comprehensive disciplines: ordinary differential equations, partial differential equations, etc.