VI. Syllabus for Qualifying Examinations

Examinations will be based on material in references listed below for each area. Some topics may not be covered in courses listed in the previous subsection, IV.(A), in which case the student should study such topics in the suggested references. A list of the principal topics in each area is presented as an overview, but not as a detailed outline of the reference material. Previous exams are available at: http://www.math.purdue.edu/academic/grad/qualexams/

1. Real Analysis (MA 544)
   1. Topics:
   1. Topology of R (open, closed, compact, connected, category).
   2. Continuity, semi-continuity, sequences of continuous functions and types of convergence, equicontinuity and compactness in C[0, 1], Stone-Weierstrass theorem.
   3. Construction of Lebesgue measure on R, abstract measure spaces, the Lebesgue integral and L^p-spaces.
   4. Differentiation: (i) Bounded variation and Helly Selection Theorem, (ii) Vitali Covering Theorem, differentiation of monotone functions, absolute continuity and the Fundamental Theorem of Calculus.
   2. Books:
   • For (a) and (b): W. Rudin, Principles of Mathematical Analysis R. Bartle, The Elements of Real Analysis Natanson, Theory of Functions of a Real Variable, v.1.
   • For (c): W. Rudin, Real and Complex Analysis A. Torchinsky, Real Variables H. L. Royden, Real Analysis
   • For (d): Natanson, Torchinsky and Royden
2. Abstract Algebra (MA 553)
   1. Prerequisites:
      • Some undergraduate level linear algebra and group theory such as is found on pages 1-99 of M. ArtinÕs Algebra and, in addition, D&F (see below), Chapters 0-3 (except ¤3.4).
   2. Topics:
      1. Group theory; Sylow theorems; Jordan-Hölder theorem; solvable groups. [D&F, Chapters 4-5, plus §3.4].
      2. Ring theory; unique factorization in polynomial rings and principal ideal domains. [D&F, Chapters 7-9].
      3. Field theory; ruler and compass constructions, roots of unity, finite fields, Galois theory; solvability of equations by radicals. [D&F, Chapters 13-14, up to and including 14.7].
   3. Book:
      • [D&F]: D. Dummit and R. Foote, Abstract Algebra, 2nd. edition.
3. Complex Analysis (MA 530)
1. Topics:
   1. Cauchy-Riemann equations; conformality and other properties of analytic functions; linear fractional transformations; special functions.
   2. Taylor and Laurent series; absolute and uniform convergence.
   3. CauchyÕs theorem, formula, residue theorem, inequality; Morera's theorem; classification of singularities; Liouville's theorem; fundamental theorem of algebra; Casorati-Weierstrass theorem; definite integrals; maximum modulus theorem; Schwarz's lemma; Rouche's theorem; Weierstrass' theorem.
   4. Harmonic functions; Poisson formula, Schwarz's theorem; harmonic conjugates; reßection principle.
2. Book:
   • Ahlfors: Complex Analysis, 3rd Edition, pp. 1-48, 67-84, 101-186.
4. Linear Algebra (MA 554)
1. Topics:
   1. (a) Vector spaces; linear maps; matrices; determinants; systems of linear equations.
   2. Inner products; hermitian, unitary and normal operators.
   3. Modules over a principal ideal domain; finitely generated abelian groups; Jordan and rational canonical forms for a linear transformation.
2. Books:
   • Hoffman and Kunze, Linear Algebra, Chapters 1-8 (omitting ¤¤5.6, 5.7)
   • Jacobson, Basic Algebra I, Chapter 3 (omit ¤3.11)
5. Differential Geometry (MA 562)
   1. Prerequisites:
      • Some undergraduate multivariate calculus and topology as found in Munkres, Chapters 1-4 (see below) including the topology of Rⁿ, the chain rule for mappings from Rⁿ into R^m, the implicit and inverse function theorems, and Jacobians.
   2. Topics:
      1. Differentiable manifolds and submanifolds; differentiable mappings, rank of a mapping and immersions, submanifolds, tangent and cotangent bundles.
      2. Vector fields, Lie groups, One parameter groups, Lie bracket, Frobenius' theorem.
      3. Tensors and tensor fields on manifolds; exterior algebra, orientation, integration on manifolds, Stokes' Theorem on manifolds.
   3. Books:
      • J. Munkres, Analysis on Manifolds, Chapters 1-4.
      • W. Boothby, Differentiable Geometry and Riemannian Geometry, Chapters 1-6.
6. Probability (MA 519)
   1. Topics:
      1. Probability spaces and axioms; countable additivity of probability laws.
      2. Combinatorial analysis.
      3. Discrete random variables.
      4. Continuous random variables.
      5. Jointly distributed random variables; distributions of functions of random variables.
      6. Expectations, variance, moments.
      7. Jointly normal random variables in detail.
      8. Limit theorems (e.g., weak law of large numbers and especially the central limit theorem).
   2. Books:
      • Hoel, Port, and Stone Introduction to Probability , Chapters 1-7, (omitting starred sections, except 5.4 and 6.7; Some of Chapter 8)
   3. Supplements to:
   • Multidimensional Changes of Variables:
     1. Stroock, A Concise Introduction to the Theory of Integration, Section IV.3
     2. Papoulis, Probability, Random Variables and Stochastic Processes, Sections 7.1 and 7.2 Jointly normal random variables: see Breiman, Probability: With a view towards applications Central Limit Theorem: see, for example, Breiman, Probability Chapter 1 and section 8.6 (pp 167-170)
7. Applied Mathematics (MA 523)
   1. Topics:
      1. Integral curves and surfaces of vector fields; Quasi-linear and linear equations of first order.
      2. Characteristics; classification; canonical forms.
      3. Separation of variables; Sturm-Liouville problems; Fourier series and convergence theorems.
      4. Equations of mathematical physics; Laplace equation; wave equation; heat equation.
      5. Cauchy-Kowalewski theorem; Holmgren Uniqueness theorem.
   2. Books:
      • Churchill and Brown, Fourier Series and Boundary Value Problems,, 4th Edition, Chapters 2, 3, 4
      • Zachmanoglou and Thoe, Introduction to Partial Differential Equations, Chapters 2, 3, 4, 5-10
      • John, Partial Differential Equations Chapter 1 : §§1-6 Chapter 2 : §§1-4 Chapter 3 : §§1-6 Chapter 4 : §§1-3 Chapter 5 : §1 Chapter 7 : §1
8. Topology (MA 571)
   1. Topics:
      1. Topological spaces and continuous functions, basis for a topology, product topology (for finite and infinite products), subspace topology, topology induced by a metric.
      2. Connectedness, path connectedness, local connectedness. Compactness and local compactness. The compact-open topology.
      3. Homotopy of paths and the fundamental group. Covering spaces. Fundamental groups of some important spaces (circle, sphere, torus, projective space).
      4. Free products of groups. Statement of the Seifert-van Kampen theorem (but not the proof). Use of Seifert-van Kampen to calculate the fundamental groups of various spaces. Classification of Surfaces.
   2. Books:
      • Munkres, Topology, Second Edition. Chapter 2 (including the starred sections and the supplementary exercises), Chapter 3 (including the starred sections but not the supplementary exercises), Section 46, Chapter 9 (not including the starred sections), Chapter 11, and Chapter 12.
9. Logic (MA 585)
   1. Topics:
      1. (a) Propositional and predicate calculus.
      2. Gödel's completeness and compactness theorems.
      3. Primitive recursive and recursive functions; G¬odel's incompleteness theorem, Tarski's and Church's theorems; undecidability.
      4. Nonstandard Models.
   2. Books:
      • Enderston, A Mathematical Introduction to Logic, Chapters 1, 2, 3
      • Mendelson, Introduction to Mathematical Logic (3rd Edition), Chapters 1, 2, 3 (omit ¤¤1.5, 1.6, 2.15)
10. Numerical Analysis (CS/MA 514)
    1. Topics:
       1. Machine Arithmetic, Error Propagation and the Conditioning of Problems: real numbers, machine numbers, rounding; machine arithmetic; propagation of rounding errors, cancellation errors; conditioning of problems, examples.
       2. Approximation and Interpolation: least squares approximation and data fitting; orthogonal polynomials; polynomial interpolation, Lagrange's formula; interpolation error and convergence; interpolation at Chebyshev points, Chebyshev polynomials; Newton's form of the interpolation polynomial; Hermite interpolation; inverse interpolation; interpolation by means of spline functions, minimal properties of spline interpolants.
       3. Numerical Differentiation and Integration: finite difference approximation of derivatives; numerical integration by composite trapezoidal and Simpson rules; Newton-Cotes formulae; Gaussian quadrature formulae; approximation of linear functionals, methods of interpolation and undetermined coefficients; extrapolation methods, Romberg integration.
       4. Nonlinear Equations: examples; iterative methods, order of convergence; bisection method; secant method and its convergence properties; Newton's method, local and global convergence; algebraic equations; systems of nonlinear equations (briefly).
       5. Ordinary Differential Equations: one-step methods, local and global error; Runge-Kutta methods; stiff equations; multistep methods.
    2. Books:
       • G. Dahlquist & A. Björck, Numerical Methods
       • J. Stoer & R. Bulirsch, Introduction to Numerical Analysis