Spring 2012

- First, it is very important to get a copy of the past written exams from Tamar's office.
- There is a Written Exam Wiki with solutions to almost all of the past written exam problems: good if you want to check your answers or find out how to do a problem on which you're stuck.
- I would advise you to find at least one good reference on each subject, from which you can review the theory. What is a good reference is a matter of personal taste, pick a book that appeals to you and that contains enough information for preparing for the exam. You can find a few suggestions below.
- There is various information that you can find on websites of previous written exam workshops:
- Fall 2011 - Behzad Mehrdad
- Spring 2011 - Evan Chou
- Fall 2009 - William Perkins
- Fall 2007 - Miranda Holmes
- Andrew Suk has written up solutions to exam problems

- Buck, Advanced Calculus
- Courant, Differential and Integral Calculus (2 Volume set)
- Courant, John, Introduction to calculus and analysis:

Volume I, Volume II/1, Volume II/2 - Ponnusamy, Foundations of Mathematical Analysis,

(fulltext available on springerlink) - Rudin, Principles of mathematical analysis

- Lax, Linear algebra and its applications
- Strang, Introduction to linear algebra
- Trefethen, Bau, Numerical linear algebra (Good for matrix decompositions and SVD)

- Ahlfors, Complex analysis
- Conway, Functions of one complex variable, Volume I, Volume II
- Lang, Complex analysis (my personal favorite)

General information, questionnaire, and start with advanced calculus:

Elementary inequalities, such as Cauchy's inequality and AM-GM inequality, Series: convergence tests for positive series such as comparison test, limit comparison test, ratio test, root test

Some standard series, Cauchy condensation test, recognizing a Riemann sum, telescoping series, (generalized) Dirichlet test, power series and Cauchy-Hadamard Theorem, approximation to the identity, methods to interchange limits and integrals such as Dominated Convergence Theorem.

Multivariable calculus: Derivatives, partial derivatives, mixed partial derivatives, optimization and Lagrange multipliers and applications to prove inequalities, Inverse and Implicit Function Theorems, Green's Theorem, Stokes Theorem, Divergence Theorem, Fourier Series

Change of basis, matrix representation of linear maps, Jordan decomposition, rank-nullity theorem.

Exercise Sheet Linear Algebra 1

Matrix Decompositions: Polar, SVD, LU, Choleski, QR, Schur.

Exercise Sheet Linear Algebra 2

Powers of matrices, positive definite matrices, characterization of commutators, Gram matrix, projections.

Exercise Sheet Linear Algebra 3

Complex numbers, complex functions, complex differentiability, holomorphic functions, conformal maps, Cauchy-Riemann equations, (formal) power series, Cauchy-Hadamard, open mapping and inverse function theorem, maximum principles, path integration, Cauchy-Goursat, primitives of holomorphic functions.

Cauchy Formula, Laurent Series, singularities, Casorati-Weierstrass, analytic automorphisms of complex numbers, Residue formula, Rouche's theorem.

Exercise Sheet Complex Variables 1

Calculation of integrals by Residue Theorem, evaluation of series by Residue Theorem, Conformal mappings, Fractional linear transformations.

Exercise Sheet Complex Variables 2More on conformal maps, Weierstrass representation theorem, Hadamard's theorem.

Exercise Sheet Complex Variables 3Analytic continuation, Schwarz reflection, Blaschke products. Work in class on exercises from exam Fall 2007.

Final remarks, some interesting exercises, finish exam Fall 2007.