Problems and Solutions in Mathematics
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.
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Part 2 Topology
Part 3 Differential Geometry
Part 4 Real Analysis
Part 5 Complex Analysis
Part 6 Partial Differential Equations
Abbreviations of Universities in this Book
abelian ac G analytic function Assume Borel bounded Cauchy problem compact subset Compute conformal map connected constant continuous function continuous map contradiction converges coordinates covering map curve defined denote differential disjoint domain easy elements equality equation exists ﬁeld finite fn(z follows formula function f Galois extension Gauss curvature Gaussian curvature geodesic harmonic Hence homology homomorphism homotopy implies Indiana inequality integral intentionally left blank Iowa irreducible isomorphic Lebesgue measure Let f limn_,oo linear manifold map f matrix metric minimal polynomial neighborhood nilpotent normal subgroup normal vector obtain Obviously open set plane positive Prove real number respect Riemannian Riemannian manifold roots satisfies sequence single-valued smooth Solution space Stanford subgroup subspace Suppose surface surjective tangent Theorem topological uniformly unique unit disk vector field Zero