Ta-Tsien L. (ed.) Problems and solutions in mathematics

2nd ed. — Singapore: World Scientific, 2011. — 804 p. — (Major American Universities Ph.D. Qualifying Questions and Solutions).Compiled by:
Chen Ji-Xiu, Jiang Guo-Ying, Pan Yang-Lian, Qin Tie-Hu, Tong Yu-Sun, Wu Quan-Shui and Xu Sheng-ZhiThis book contains a selection of more than 500 mathematical problems and their solutions from the PhD qualifying examination papers of more than ten famous American universities. The mathematical problems cover six aspects of graduate school mathematics: Algebra, Topology, Differential Geometry, Real Analysis, Complex Analysis and Partial Differential Equations. While the depth of knowledge involved is not beyond the contents of the textbooks for graduate students, discovering the solution of the problems requires a deep understanding of the mathematical principles plus skilled techniques. For students, this book is a valuable complement to textbooks. Whereas for lecturers teaching graduate school mathematics, it is a helpful reference.Algebra:
Linear Algebra
Group Theory
Ring Theory
Field and Galois TheoryTopology:
Point Set Topology
Homotopy Theory
Homology TheoryDifferential Geometry:
Differential Geometry of Curves
Differential Geometry of Surfaces
Differential Geometry of ManifoldReal Analysis:
Measurability and Measure
Integral
Space of Integrable Functions
Differential
Miscellaneous ProblemsComplex Analysis:
Analytic and Harmonic Functions
Geometry of Analytic Functions
Complex Integration
The Maximum Modulus and Argument Principles
Series and Normal FamiliesPartial Differential Equations:
General Theory
Elliptic Equations
Parabolic Equations
Hyperbolic Equations