Integration and Complex Variables
Topics covered on the exam include the following. Most of these topics are discussed in Math 602.
 Construction and properties of the Lebesgue measure
 Lebesgue measurable and integrable functions
 The dominated convergence theorem, Fatou's lemma, the monotone convergence theorem, and the bounded convergence theorem
 Analytic functions, Taylor series, Cauchy's theorem, the generalized Cauchy integral formula
 The maximum modulus principle and Liouville's theorem
 Laurent series, the residue theorem and applications to computation of integrals
References are:
 [1], Chapter 10 and [2] Chapters 24 and 7 for items 13.
 [1] Chapter 16 and [3] Chapters 2, 4, and 5 for items 46.
[1] Tom Apostol,
Mathematical Analysis, 2nd edition, Addison Wesley, 1974.
[2] H.L. Royden, P.M. Fitzpatrick.
Real Analysis, 4th edition. Person, 2010.
[3] L. Ahlfors,
Complex analysis, 3rd edition. McGrawHill, 1979
Applied Mathematics
Topics covered on the exam include the following. Most of these topics are discussed in Math 617.
 Ordinary Differential Equations
 Fourier series (use in PDEs) [1] ch. 3.
 SturmLiouville theory (as preparation for eigenfunction expansions of PDEs) [1] ch. 5.
 Green's functions [2] §5.5
 Bessel and Legendre functions (as eigenfunctions for PDEs in alternate geometries) [1] ch. 7.
 SecondOrder Linear PDEs
 Separation of variables [1], ch. 2
 Fourier and Laplace transform methods [1] ch. 10 and 13
 The diffusion equation (maximum principle) [1] ch. 2
 Laplace's equation (maximum principle, Poisson's integral formula): [1] §9.5
 The wave equation (characteristics, d'Alembert's solution): [1] §12.3
 Variational principles: [2] §4.1, 4.3
Section and chapter references are given from
[1] Haberman, Richard.
Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th ed. New York: Pearson, 2013.
[2] Logan, J. David.
Applied Mathematics, 4th ed. New York: Wiley, 2013.
Numerical Methods
Topics covered on the exam include the following. Most of these topics are discussed in Math 611.

Polynomial interpolation:

Lagrange and Newton form [QSS, 8.1,8.2].

Piecewise interpolation in 1 and 2D [QSS, 8.4, 8.6].

Splines [8.7,8.8].

Numerical Integration and Finite Differences:

NewtonCotes and composite formulae [9.2, 9.3, 9.4].

Singular integrals [9.8].

Orthogonal polynomials [10.1].

Gaussian quadrature [10.2, 10.4].

Approximation of derivatives [10.10]

Numerical solution of Ordinary Differential Equations:

Gronwall Lemma [11.1].

One step methods [11.2],

Stability and consistency [11.3]

Difference equations [11.4]

Multistep methods [11.5]

Consistency and stability [11.6]

Runge Kutta methods [11.8]

Stiff Problems [11.10]

Finite Differences

Finite difference approximation to two point boundary value problems [QSS12.2]

Discretization of the heat equation [13.2]
The section numbers refer to
Numerical Mathematics by A. Quarteroni, R. Sacco, and F. Saleri.
Algebra
Topics covered on the exam include the following. Most of these topics are discussed in Math 650.
 Elementary Group Theory
 basics (group, subgroup, cosets, Theorem of Lagrange) (Ch. 1,2)
 Homomorphisms, normal subgroups and Isomorphism Theorems. (Ch. 3)
 Classification of Finite Abelian Groups. (Ch. 7AB)
 Commutative Rings
 Polynomial rings, PIDs and UFDs. (Ch. 16)
 Fields and field extensions, splitting fields. (Ch. 17)
 Finite Fields (Ch. 21AB)
 Structure and uniqueness. Subfields. (Ch. 21A)
 Irreducible polynomials (and counting them). (Ch. 21B)
Chapter numbers refer to I.M. Isaacs,
Algebra, A graduate course.
Stochastic Processes
Topics covered on the exam include the following. Most of these topics are discussed in Math 631.
 Markov chains (discretetime) (Chapter 1)
 Continuoustime Markov chain (Chapter 4)
 Martingales (discretetime) (Chapter 5)
 Key examples:
 Random walks
 Birth and death processes
 Branching processes
 Markov chain Monte Carlo
Chapter numbers refer to R. Durrett,
Essentials of Stochastic Processes