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ISBN 10: 9814304964
ISBN 13: 9789814304955
Author: Ta Tsien Li
This 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.
Part 1 Algebra
Section 1 Linear Algebra
Section 2 Group Theory
Section 3 Ring Theory
Section 4 Field and Galois Theory
Part 2 Topology
Section 1 Point Set Topology
Section 2 Homotopy Theory
Section 3 Homology Theory
Part 3 Differential Geometry
Section 1 Differential Geometry of Curves
Section 2 Differential Geometry of Surfaces
Section 3 Differential Geometry of Manifolds
Part 4 Real Analysis
Section 1 Measurability and Measure
Section 2 Integral
Section 3 Space of Integrable Functions
Section 4 Differential
Section 5 Miscellaneous Problems
Part 5 Complex Analysis
Section 1 Analytic and Harmonic Functions
Section 2 Geometry of Analytic Functions
Section 3 Complex Integration
Section 4 The Maximum Modulus and Argument Principles
Section 5 Series and Normal Families
Part 6 Partial Differential Equations
Section 1 General Theory
Section 2 Elliptic Equations
Section 3 Parabolic Equations
Section 4 Hyperbolic Equations
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