Syllabus for the PhD Preliminary Examination in Analysis
Metric spaces: open and closed sets, convergence, compactness, connectedness, completeness, continuity, uniform continuity, uniform convergence, equicontinuity and the Ascoli-Arzela Theorem, contraction mapping theorem
Single variable analysis: numerical sequences and series, differentiation, mean value theorem, Taylor's theorem, function series and power series, uniform convergence and differentiability, Weierstrass approximation theorem, Riemann integral, sets of measure zero
Several variables analysis: differentiability, partial derivatives, inverse and implicit function theorems, iterated integrals, Jacobians, change of variable in multiple integrals
Vector analysis: Stokes theorem, Green's theorem, divergence theorem
- Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, Freeman, 1993
- W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976
- Michael Spivak, Calculus on Manifolds, Addison-Wesley, 1965
- T. Apostol, Mathematical Analysis, Addison-Wesley, 1974
The analysis syllabus is downloadable as a PDF file.
Syllabus for the PhD Preliminary Examination in Linear Algebra
Vector spaces: subspaces, linear independence, bases, dimension, isomorphism, linear functionals, dual space, adjoints, inverses and reducibility
Matrices and linear transformations: range, kernel, determinants, isomorphisms, change of basis, eigenvalues, eigenvectors, minimax Theory of eigenvalues, Gersgorin discs, minimal polynomial,
Cayley-Hamilton theorem, similarity, polar and singular value decomposition, spectral theorem, Jordan cannonical forms. Hermitian, symmetric, and positive definite matricies. Matrix andvector norms, convergence of sequences, powers etc. of matrices
Inner product spaces: inner products, norms, orthogonality, projections, orthogonal complement, orthonormal basis, Gram-Schmidt orthogonalization, linear functionals, isometries, normal operators, spectral theory, basic inequalities such as Cauchy Schwarz
- P.R. Halmos, Finite Dimensional Vector Spaces, Springer, 1993
- R. Horn and C. Johnson, Matrix Analysis, Cambridge, 1999
- Peter Lax, Linear Algebra, Wiley-Interscience, 1997.
- K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice Hall, 1972
- P. N. de Souza and J.-G. Silva, Berkeley Problems in Mathematics, Springer,
- Berlin, 2004, (Chapter 7: Linear Algebra)
The linear algebra syllabus is downloadable as a PDF file.