## Problems and Solutions in MathematicsThis 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 problems cover six aspects of graduate school mathematics: Algebra, Differential Geometry, Topology, Real Analysis, Complex Analysis and Partial Differential Equations. The depth of knowledge involved is not beyond the contents of the textbooks for graduate students, while solution of the problems requires deep understanding of the mathematical principles and skilled techniques. For students this book is a valuable complement to textbooks; for lecturers teaching graduate school mathematics, a helpful reference. |

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### Contents

Linear Algebra | 3 |

Group Theory 26 | 26 |

Ring Theory 44 | 44 |

Field and Galois Theory 59 | 59 |

Point Set Topology 83 | 83 |

Homotopy Theory 99 | 99 |

Homology Theory 118 | 118 |

Differential Geometry of Curves 153 | 153 |

Differential 302 | 302 |

Miscellaneous Problems 322 | 322 |

Analytic and Harmonic Functions 335 | 335 |

Geometry of Analytic Functions 360 | 360 |

Complex Integration 377 | 377 |

The Maximum Modulus and Argument Principles 413 | 413 |

Series and Normal Families 433 | 433 |

General Theory 457 | 457 |

Differential Geometry of Surfaces 171 | 171 |

Differential Geometry of Manifold 194 | 194 |

Measurablity and Measure 231 | 231 |

Integral 256 | 256 |

Space of Integrable Functions 283 | 283 |

Elliptic Equations 472 | 472 |

Parabolic Equations 496 | 496 |

Hyperbolic Equations 513 | 513 |

Abbreviations of Universities in This Book 539 | 539 |

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abelian analytic function Assume boundary bounded Cauchy problem circle closed compact subset Compute conformal map conjugate constant continuous function continuous map contradiction converges coordinates Courant Inst covering map curve defined deformation retract denote diffeomorphism differential disjoint domain easy elements entire function equal equation equivalence exists finite follows formula Galois extension Gauss curvature geodesic harmonic function Hausdorff Hence homology homomorphism homotopy implies Indiana Indiana-Purdue inequality integral Iowa irreducible polynomial isometry isomorphic J-oo Jn Jn Lebesgue measure lemma Let f(z Let G linear matrix Mayer-Vietoris sequence measurable set metric minimal polynomial nilpotent obtain Obviously open set path connected plane Prove radius real number respect Riemannian manifold roots satisfies Show single-valued branch smooth Solution splitting field Stanford submanifold subspace Suppose surface surjective tangent theorem transformation uniformly unique unit disk universal covering vector field zero