1. Arithmetic ability: Advanced mathematics involves complicated calculation and deduction, which requires students to have a solid arithmetic foundation, including addition, subtraction, multiplication and division, fractions, decimals, percentages and other operations.

2. Algebraic ability: Algebraic symbols and expressions are widely used in derivation and calculation in advanced mathematics. Students need to master the rules of algebraic operation, including polynomial expansion, factorization and solving equations.

3. Geometric ability: Many concepts and theorems in advanced mathematics are related to geometry. Students need to have some knowledge of geometry, including the basic concepts, properties and theorems of plane geometry and solid geometry.

4. Function and ultimate ability: The function in advanced mathematics is the basis of studying mathematical problems. Students need to understand the definition, properties and images of functions, and be able to operate and transform functions. In addition, limit is an important concept in higher mathematics, and students need to master the definition, nature and calculation method of limit.

5. Calculus ability: Calculus is one of the core contents of advanced mathematics. Students need to understand the concepts of derivative and integral, and be able to perform derivative and integral operations of functions. In addition, students need to master the basic concepts and solutions of differential equations.

6. Matrix and linear algebra ability: Matrix and linear algebra in advanced mathematics are tools for studying vector space and linear transformation. Students need to understand the basic operations and properties of matrices, and be able to solve matrices and linear equations.

7. Probability and statistics ability: Probability theory and mathematical statistics in advanced mathematics are methods to study random phenomena and data analysis. Students need to understand the basic concepts, properties and calculation methods of probability statistics.

Learning advanced mathematics requires students to have good logical thinking ability, problem analysis ability, abstract thinking ability and problem solving ability in addition to the above basic mathematical abilities. These abilities can be cultivated and improved through a lot of practice and practice.