# Preliminary Examination Syllabi

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## First Tier

#### Linear Algebra

Topics covered on the exam include the following. Most of these topics are discussed in Math 672.

- Subspaces, bases and dimension (Chapters 1 - 2, [1], ).
- Linear transformations and matrix representations (Chapter 3, [1]; Chapters III - IV, [3]).
- Determinants and rank (Chapter VI, Section V.3, [3]).
- Inner products and inner product spaces (Chapter 6, [1]).
- Linear functionals, adjoints, and dual spaces (Chapter 7, [1]; Chapter I. F, [2]).
- Bilinear forms, Hermitian forms, and Quadratic forms ( Sections IV.A - IV.C, [2], Chapter V [3]).
- Eigenvalues, eigenvectors, and characteristic polynomials (Sections VIII.1 - VIII.2, [3]).
- Cayley-Hamilton Theorem (Sections III.A - III.C, [2]).
- Operators on inner product spaces and Spectral Theorems (Chapter 7, [1]; Section III.D, [2]; Section VIII.3 - VIII.6, [3]).
- Jordan Canonical Form (Chapter XI, [3]; Section III.E, [2]; Chapter 8, [1])

Chapter and section numbers refer to

[1] S. Axler,

Linear Algebra Done Right, Second Edition, Springer-Verlag, 1997.[2] M.L. Curtis,

Abstract Linear Algebra, Springer-Verlag, 1990.[3] S. Lang,

Linear Algebra,Third Edition, Springer-Verlag, 1987.

#### Real Analysis

Topics covered on the exam include the following. Except the part on multivariable calculus, most of these topics are discussed in Math 600.

- Metric Spaces: open and closed sets, compactness, connected sets, complete sets, continuous functions on metric spaces ([1], Chapters 3 and 4, [3] Chapter 2).
- Continuity and Differentiation: mean value theorem, Rolle's theorem, Taylor's formula, derivatives of vector valued functions, uniform continuity, monotonic functions ([1], Chapters 5 and 6, [3] Chapter 4).
- Infinite Sequences and Series: Limit superior and limit inferior, monotonic sequences, alternating series, absolute and conditional convergence, power series, tests for convergence of series, rearrangement of series ([1], Chapter 8, [3] Chapter 3).
- Sequences of Functions: Pointwise convergence, uniform convergence, uniform convergence and continuity, differentiability and integration ([1], Chapter 9, [3] Chapter 7).
- Riemann integration ([1] Chapter 7, [3] Chapter 6).
- Functions of Several Variables: Directional derivatives, the total derivative, Jacobians, inverse function theorem, implicit function theorem, extrema problems ([1], Chapters 12 and 13).
- Vector Calculus: Line integrals, Green's theorem, surface integrals, Stokes theorem, the divergence theorem ([2], Chapters 10, 11 and 12).

Chapter numbers refer to

[1] T. Apostol, *Mathematical Analysis*, 2nd edition, Addison Wesley, 1974.[2] T. Apostol, *Calculus*, Vol. 2, 2nd edition, John Wiley, 1969.[3] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw Hill, 1976.

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## Second Tier

#### 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 2-4 and 7 for items 1-3.
- [1] Chapter 16 and [3] Chapters 2, 4, and 5 for items 4-6.

[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. McGraw-Hill, 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.
- Sturm-Liouville 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.
- Second-Order 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, 8.3,8.5].
- 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.

#### 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 (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*

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