Real variable function: Lebesgue measure on r n: the concept and basic properties of measurable function; Integrals of measurable functions and their Lebesgue integrals: control convergence theorem of integrals, Levi lemma and Fatou lemma; Product measure and Fourier theorem; Monotone function, bounded variation function and fully continuous function.

Complex variable function: differentiability and analysis, cauchy-riemann equations, Cauchy integral theorem, Cauchy integral formula, maximum modulus principle, Schwartz lemma, uniqueness theorem of analytic function, harmonic function, power series and Laurent series, isolated singularity, residue and its application.

Abstract algebra: group: what is the decomposition of group, subgroup and coset, the concepts of cyclic group, normal subgroup and quotient group, the basic theorem of homomorphism, permutation group and the role of group in set. Ring and field: basic concepts, ring homomorphism (definition, ideal, quotient ring, first isomorphism theorem, prime ring and prime field, China remainder theorem, prime ideal and maximal ideal), concepts and main examples of unique factorization whole ring and Euclidean whole ring, polynomial ring over field, simple algebraic extension of field, preliminary knowledge of finite field. Basic requirements: the emphasis is on the understanding of basic concepts and their important examples, knowing the most important theorems and their simple applications, and the requirements for problem-solving skills are not high.

Differential geometry: curve theory in three-dimensional Euclidean space, including curvature, torsion and the basic theorem of curve theory; The basic theory of three-dimensional Euclidean space surface includes the first basic form, the second basic form, principal curvature, average curvature and Gaussian curvature.

I see from Chinese that there is no bibliography about the scope of the exam. I hope it helps you.