Oral Qualifying Exam

The oral qualifying exam is a two-hour examination covering two graduate-level topics selected from the seven graduate level topics listed below. It is given by a committee consisting of two members of the graduate faculty. The purpose of the exam is to determine whether a student has mastered the relevant beginning-level graduate coursework to a point where they can continue on to advanced-level coursework and research leading to their Ph.D. The oral qualifying exam must be passed before the beginning of the spring semester of the second year of study.

Process

  1. Select two topics from the seven graduate level topics listed below.
  2. You then contact the Director of Graduate Studies to have them appoint an exam committee. If you have a preference, you may request faculty members.
  3. The DGS will let you know which faculty members have agreed to sit on the exam committee. You can then contact those faculty to work out the date and time for your exam, which usually takes place in a faculty office.
    NOTE: When you talk to the professors who will give you the exam, make sure you understand what they expect you to know.
  4. When a date and time is confirmed for the exam, contact the DGS Assistant to give them that information AT LEAST ONE WEEK before the exam date.
    NOTE: It is recommended that you email your examiners a day or two before the exam to confirm their attendance.
  5. There are several possible outcomes of the oral qualifying exam:

    1. Passing both topics at the Ph.D. level. The qualifying exam requirements are then complete.
    2. Falling short of passing one or both topics at the Ph.D. level will be categorized into two levels:
      1. Passing at the master's level.
      2. Failing the exam entirely.
    3. The committee may decide to allow students who do not pass a topic at the Ph.D. level to retake the exam (in front of the full committee) once more.
    4. Students who ultimately do not pass the oral qualifying exams at Ph.D. level will not continue in the Ph.D. program.
  6. Decisions of the committee will be entered on the Math Dept Oral Qual Form.

Graduate Level Topics

Each tab below describes the syllabus topics and suggested reading for each. The amount of material covered on each topic corresponds roughly to the content of a one-semester introductory graduate level course. Many typical oral exam questions probe the student's ability to apply general theory to specific examples.

Algebra Syllabus

For the Oral Qualifying Exam

Groups

  • Elementary concepts (homomorphism, subgroup, coset, normal subgroup), solvable groups, commutator subgroup, Sylow theorems, structure of finitely generated Abelian groups.
  • Symmetric, alternating, dihedral, and general linear groups.

Rings

  • Commutative rings and ideals (principal, prime, maximal).
  • Integral domains, Euclidean domains, principal ideal domains, polynomial rings, Eisenstein's irreduciblility criterion, Chinese remainder theorem.
  • Structure of finitely generated modules over a prinicpal ideal domain.

Fields

  • Extensions: finite, algebraic, separable, inseparable, transcendental, splitting field of a polynomial, primitive element theorem, algebraic closure.
  • Finite Galois extensions and the Galois correspondence between subgroups of the Galois group and subextensions.
  • Solvable extensions and solving equations by radicals.
  • Finite fields.

References

  • M. Artin, Algebra
  • Dummit and Foote, Algebra
  • S. Lang, Algebra
  • T.W. Hungerford, Algebra

Complex Analysis Syllabus

For the Oral Qualifying Exam

  • Complex differentiation, Cauchy-Riemann equations, power series, exponential and trigometric functions.
  • Cauchy's theorem and integral formula, Cauchy's inequalities, Liouville's theorem, Morera's theorem, classification of isolated singularities, Taylor series, meromorphic functions, Laurent series, fundamental theorem of algebra, residues, winding numbers, argument principle, Rouch\'e's theorem, local behaviour of analytic mappings, open mapping theorem.
  • Harmonic functions, maximum principle, Poisson integral formula, mean value property.
  • Conformal mappings, linear fractional transformations, Schwarz lemma.
  • Infinite products, analytic continuation, multi-valued functions, Schwarz reflection principle, monodromy theorem.
  • Statement and consequences of Riemann mapping theorem and Picard's theorem.

References

  • L. Ahlfors, Complex Analysis
  • J.B. Conway, Functions of One Complex Variable
  • R. Churchill, Complex Variables and Applications
  • S. Lang, Complex Analysis
  • Levinson and Redheffer, Complex Variables
  • K. Knopp, Theory of Functions, vols I-III.

Differential Equations Topic List

For the Oral Qualifying Exam

For the oral qualifying exam in Differential Equations the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one-semester graduate course which has no other graduate course as a prerequisite. Once you have chosen your list discuss it with the professsor who will examine you.

Ordinary Differential Equations

  1. Fundamental existence theorems; uniqueness with the Lipschitz condition; the Gronwall inequality; continuation of solutions to the boundary; dependence on parameters and variational equations.
  2. Solution of linear systems of equations with constant coefficients; the exponent of a matrix; the Jordan canonical form and implications for the large time behavior of solutions; classification and phase portraits of 2 by 2 linear systems.
  3. The simplest numerical methods, order of accuracy.
  4. Equilibria; notion of stable and asymptotically stable equilibria; linearization about an equilibrium; stability of an equilibrium as a consequence of linearized stability; Liapunov functions and their implications for stability.
  5. The phase plane; limit cycles; the van der Pol equation; the Poincare-Bendixson Theorem (statement, not proof); the phase portrait of the pendulum equation; strange attractors and the Lorenz system; chaos; bifurcation of equilibria; Hopf bifurcation.

Partial Differential Equations

  1. Notion of well-posed problem; the classical examples (wave equation, heat equation, Laplace's equation); solution by Fourier series and Fourier transform.
  2. First-order equations; geometric interpretation of solutions; method of characteristics for linear and quasilinear equations; domain of dependence and influence; the simplest numerical methods for first-order linear hyperbolic equations; numerical stability and the CFL condition.
  3. The wave equation in one space dimension, explicit solution and energy conservation; solution in 3-D by spherical means and 2-D by descent; domain of dependence and influence for the wave equation.
  4. The heat equation in free space, fundamental solution, smoothing property; notion of similarity solutions; maximum principle; Duhamel's principle for nonhomogeneous problems.
  5. Two-point boundary value problems on an interval, and their Green's functions; Laplace's equation and Poisson's equation; the maximum principle and mean value property for harmonic functions; fundamental solutions of Laplace's equation; representation of solutions by boundary integrals; Dirichlet and Neumann problems; Green's functions (definition, half-space, disk).
  6. Notion of distributions, especially the delta function; distributional interpretation of fundamental solutions; weak derivatives; weak formulation of the Dirichlet problem in Hilbert space; eigenvalues of Laplacian in a bounded domain; solution of wave equation or heat equation in a bounded domain by eigenfunction expansions.

Topology and/or Differential Geometry Topic List

For the Oral Qualifying Exam

For the oral qualifying exam in Topology and/or Differential Geometry the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one-semester graduate course. Once you have made your selections discuss them with the professor who will examine you.

Topology

  • Basic topological notions: path connectivity, connectivity, product topology, quotient topology.
  • The fundamental group, computation of the fundamental group, van Kampen's theorem, covering spaces.
  • Homology: singular chains, chain complexes, homotopy invariance, relationship between the first homology and the fundamental group, relative homology, the long exact sequence of relative homology, the Mayer-Vietoris sequence, applications to computing the homology of surfaces, projective spaces, etc.
  • Topological manifolds, differentiable manifolds.

Differential Geometry of Curves and Surfaces in Euclidean Space

  • The orthogonal group in 2 and 3 dimensions, the Serret-Frenet frame of a space curve.
  • The Gauss map and the Weingarten equation for a surface in Euclidean 3-space, the Gauss curvature equation and the Codazzi-Mainardi equation for a surface in Euclidean 3-space.
  • The surfaces in Euclidean 3-space of zero Gauss curvature.
  • The fundamental existence and rigidity theorem for surfaces in Euclidean space.
  • The Gauss-Bonnet formula for surfaces in Euclidean 3-space.

Differential Geometry of Riemannian Manifolds

  • Riemannian metrics and connections.
  • Geodesics and the first and second variational formulas.
  • Completeness and the Hopf-Rinow theorem.
  • The Riemann curvature tensor, sectional curvature, Ricci curvature, and scalar curvature.
  • The theorems of Hadamard and Bonnet-Myers.
  • The Jacobi equation.
  • The geometry of submanifolds – the second fundamental form, equations of Gauss, Ricci, and Codazzi.
  • Spaces of constant curvature.

References

  • Harper and Greenberg, Algebraic Topology, a First Course, Parts I and II
  • M. do Carmo, Differential Geometry of Curves and Surfaces
  • M. do Carmo, Riemannian Geometry
  • M. Spivak, A Comprehensive Introduction to Differential Geometry
  • S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces
  • S. Sternberg, Lectures on Differential Geometry, 2nd ed.

Probability & Stochastic Processes Syllabus

For the Oral Qualifying Exam

Undergraduate Material

It is expected that the candidate knows material from a standard undergraduate post-calculus level course in probability:

  • Basic properties of probability and conditional probability including Bayes rule.
  • Discrete probability densities (binomial, Poisson, geometric, hypergeometric).
  • Continuous probability densities (normal, exponential, uniform).
  • Joint, marginal, and conditional densities.
  • Expectation, variance, standard deviation, covariance.
  • Poisson approximation to binomial.
  • Chebyshev's inequality and weak law of large numbers.
  • Central limit theorem.

Graduate Material

For the exam, the student can choose one of two tracks:

  • Track I – is only for students who will not write their thesis in probability; consists primarily of stochastic processes from a non-measure theoretic perspective and corresponds to MATH 541.
  • Track II – consists primarily of measure theoretic probability as taught in STA 711 and/or MATH 641.
Core Material (required for either Track)
  • Measure theoretic foundations of probability theory: probability spaces; random variables as measurable functions; notions of convergence (almost sure versus in probability).
  • Finite Markov chains in discrete time (recurrence vs. transience, periodicity, convergence to stationary distribution).
Track I
  • Markov chains with infinite state space: positive recurrence, null recurrence, and transience; reversible Markov chains; relationship between eigenvalues and rates of convergence to equilibrium; branching processes and random walks as examples.
  • Poisson processes: definitions, thinning, superposition, conditioning.
  • Markov chains with continuous time: infinitesimal generator; Kolmogorov equations for transition probabilities; relationship to embedded discrete time Markov chains. Birth and death chains and Markovian queues as examples.
  • Brownian motion – definition and basic properties.
Track II
  • Integration: Fatou's lemma, monotone and dominated convergence; product measures, Fubini's theorem.
  • Probabilistic measure theory: Borel-Cantelli Lemmas; pi-lambda theorem, conditions for independence of events, random variables and sigma-fields; Kolmogorov extension theorem; Zero-One Laws (Kolmogorov and Hewitt-Savage).
  • Weak and strong laws of large numbers (proofs for finite variance); law of iterated logarithm (without proof).
  • Weak convergence of probability measures; characteristic functions of random variables and their relationship to weak convergence. Central limit theorem: be able to explain the ideas that underlie the proof for iid sequences.
  • Conditional expectation; Martingales (in discrete time); upcrossing inequality, martingale convergence theoresm; Doob's inequality, Lp maximal inequality; uniform integrability; optional stopping theorem; applications to branching processes, Polya urns, Radon-Nikodym derivatives, etc.
  • Birkhoff ergodic theorem (without proof), Kac's recurrence theorem.
  • Definition of Brownian motion; Kolmogorov continuity theorem; non-differentiability of paths; strong Markov property; reflection principle; Donsker's theorem.

Notes

The student is allowed to exclude topics they are not comfortable with. However, as in Olympic diving, the score on the qualifying exam will reflect both the difficulty of the material attempted and the quality of the performance.

A good way to review and broaden your knowledge to read further on the subject.

References

Core Material

  • J. Pitman, Probability

Track I

  • R. Durrett, Essentials of Stochastic Processes
  • G. Lawler, Introduction to Stochastic Processes.
  • Grimmett and Stirzaker, Stochastic Processes.
  • T. Liggett, Continuous Time Stochastic Processes. AMS.

Track II

  • R. Durrett, Probability Theory and Examples
  • J. Rosenberg, A First Look at Rigorous Probability Theory
  • D. Khoshnevisan, Probability. AMS
  • Athreya and Lahti, Measure Theoretic Probability Theory. Springer
  • Fristedt and Gray, A Modern Approach to Probability Theory. Birkhauser
  • O. Kallenberg, Foundations of Modern Probability Theory

Real Analysis Syllabus

For the Oral Qualifying Exam

  • Outer measure, measurable sets, sigma-algebras, Borel sets, measurable functions, the Cantor set and function, nonmeasurable sets.
  • Lebesgue integration, Fatou's Lemma, the Monotone Convergence Theorem, the Lebesgue Dominated Convergence Theorem, convergence in measure.
  • LΡ spaces, Hoelder and Minkowski inequalities, completeness, dual spaces.
  • Abstract measure spaces and integration, signed measures, the Hahn decomposition, the Radon-Nikodym Theorem, the Lebesgue Decomposition Theorem.
  • Product measures, the Fubini and Tonelli Theorems, Lebesgue measure on real n-space.
  • Equicontinuous families, the Ascoli-Arzela Theorem.
  • Hilbert spaces, orthogonal complements, representation of linear functionals, orthonormal bases.

References

  • H.L. Royden, Real Analysis, Chapters 1-7, 11-12.
  • Reed and Simon, Methods of Mathematical Physics I: Functional Analysis, Chapters 1-2.
  • G.B. Folland, Real Analysis, Chapters 0-3, 6.

Scientific Computing Syllabus

For the Oral Qualifying Exam

For the oral qualifying exam in Scientific Computing the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one semester graduate course. Once you have made your selections discuss them with the professor who will examine you.

Hardware/Programming Issues
  • Machine numbers, floating point arithmetic, accumulation of rounding errors, memory hierarchy, arrays in C and FORTRAN, C++ scope, C++ classes, organization of loops for computational efficiency.
Computational Linear Algebra
  • Basic linear algebra, solution of linear equations: direct and iterative methods, convergence, matrix factorizations (LU, LL^T, QR, SVD), linear equations and least squares, eigenvalues and eigenvectors.
Iterative Methods for Nonlinear Equations
  • Fixed point theorems, Convergence proofs, linear iteration methods, Newton and secant methods for scalar equations, techniques for enhancing global convergence, Newton and quasi-Newton methods for nonlinear systems.
Approximation Theory and Interpolation
  • Interpolating polynomials, Lagrange and Newton interpolation, divided differences, piecewise polynomial approximation, least squares polynomial approximation, orthogonal decompositions: Fourier series/transforms and orthogonal polynomials.
Differentiation and Integration
  • Divided differences, Richardson extrapolation, midpoint and trapezoidal rules, the Euler-Maclaurin formula, Gaussian quadrature, singular integrals.
Initial Value Problems for Ordinary Differential Equations
  • Finite difference methods: order of accuracy, stability analysis, convergence results, Euler's explicit and implicit methods, local truncation errors/rounding errors/accumulated errors, higher order methods: Adams Bashforth and Adams Moulton methods, Runge-Kutta methods, backward differentiation formulas, stiffness.
Boundary Value Problems for Ordinary Differential Equations
  • Shooting methods, finite difference methods, finite element methods, eigenvalue problems.

References

  • K.E. Atkinson, An Introduction to Numerical Analysis, 2nd ed. (Wiley, 1989)
  • Isaacson and Keller, Analysis of Numerical Methods (Dover, 1994)
  • Kincaid, Cheney and Cheney, Numerical Analysis: Mathematics of Scientific Computing
  • Stoer and Bulirsch, Introduction to Numerical Analysis
  • J. Trangenstein, Scientific Computing