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
- , Chapter 10 and  Chapters 2-4 and 7 for items 1-3.
-  Chapter 16 and  Chapters 2, 4, and 5 for items 4-6.
 Tom Apostol, Mathematical Analysis, 2nd edition, Addison Wesley, 1974.
 H.L. Royden, P.M. Fitzpatrick. Real Analysis, 4th edition. Person, 2010.
 L. Ahlfors, Complex analysis, 3rd edition. McGraw-Hill, 1979
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)  ch. 3.
- Sturm-Liouville theory (as preparation for eigenfunction expansions of PDEs)  ch. 5.
- Green's functions  §5.5
- Bessel and Legendre functions (as eigenfunctions for PDEs in alternate geometries)  ch. 7.
- Second-Order Linear PDEs
- Separation of variables , ch. 2
- Fourier and Laplace transform methods  ch. 10 and 13
- The diffusion equation (maximum principle)  ch. 2
- Laplace's equation (maximum principle, Poisson's integral formula):  §9.5
- The wave equation (characteristics, d'Alembert's solution):  §12.3
- Variational principles:  §4.1, 4.3
Section and chapter references are given from
 Haberman, Richard. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th ed. New York: Pearson, 2013.
 Logan, J. David. Applied Mathematics, 4th ed. New York: Wiley, 2013.
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, 8.3,8.5].
- Splines [8.6,8.7].
- Numerical Integration and Finite Differences:
- Newton-Cotes 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.
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.
Topics covered on the exam include the following. Most of these topics are discussed in Math 631.
- Markov chains (discrete-time) (Chapter 1)
- Continuous-time Markov chain (Chapter 4)
- Martingales (discrete-time) (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