Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

A Guide to Topics for the Qualifying Examinations

The following describes the format and scope of Qualifying Exams in each of the six areas of graduate study. It is department policy that qualifiers be based on curriculum from the first year graduate sequences and any undergraduate prerequisites. Students, who have mastered those courses, should be able to pass the exams. Faculty members, who write the exams, are expected to implement this policy, and to adhere conscientiously to the guidelines that follow. Students, in turn, are expected to interpret each exam problem in a reasonable fashion, so as not to trivialize any solution. Copies of past exams and a record of previous passing scores are available from the department by request.

Qualifying Exams (affectionately known as Quals) are given twice a year and typically take place the week or two before classes begin each semester. A precise schedule is posted months in advance. Students are allowed six hours to take the exam. Food can be brought in to help fuel the brain. Faculty, who grade the exams, are expected to release the results before the last date for students to drop or withdraw from courses without receiving a DR or W on their transcripts, and within two weeks in any case.

The books listed for each area below should be more than sufficient to cover topics that will appear on the exam. It should be emphasized, however, that the exams are intended to test general knowledge and competence rather than any particular set of books or courses.

List of Exams

  1. ALGEBRA
  2. ANALYSIS
  3. APPLIED MATHEMATICS
  4. COMPUTATIONAL MATHEMATICS
  5. GEOMETRY/TOPOLOGY
  6. LOGIC

 

ALGEBRA

The Algebra Qualifying Exam consists of five problems, all of which are to be attempted. There will be one problem in each of: Group Theory, Ring Theory, Field Theory, and Linear Algebra. The fifth problem usually combines several of these topics. The exam is based on material usually covered in undergraduate abstract algebra, linear algebra, and first year graduate algebra (741-742). (Math 541-542 are prerequisites for 741-742.) A thorough knowledge of most of the items below should be sufficient to pass.

Group Theory
Lagrange’s theorem. Action of groups on sets. Sylow theorems. Elementary properties of p-groups, nilpotent groups and solvable groups. Symmetric and alternating groups. Elementary properties of infinite groups.
 
Ring Theory
  1. Noncommutative rings--Artinian rings. Wedderburn theorems. Chain conditions for modules. Composition series and the Jordan-Hölder theorem for operator groups. Jacobson radical. Primitive rings and the Density theorem. Zorn’s lemma.
  2. Commutative rings--Prime ideals. PID's and UFD's. Noetherian rings, Hilbert Basis theorem and the Lasker-Noether theorem. Algebraic integers and basic properties of Dedekind domains. Modules over PID's and the fundamental theorem of abelian groups. Hilbert Nullstellensatz.
Field Theory
Algebraic extensions. Splitting fields. Separability. Galois extensions and the fundamental theorem of Galois theory. Primitive element theorem. Solvable polynomials. Cyclotomic polynomials. Geometric constructions. Algebraic closures. Purely inseparable extensions. Finite division rings.
 
Linear Algebra
Vector spaces. Linear transformations and matrices. Eigenvalues and eigenvectors. Jordan and rational canonical forms. Bilinear and quadratic forms. Dual spaces. Inner products.

References:

Artin, Galois Theory
Hoffman and Kunze, Linear Algebra
Isaacs, Algebra: A Graduate Course (skip Ch.15 & 25)
Hungerford, Algebra (skip Ch.10)
Rotman, Group Theory (Ch. 1-8)

 

Analysis

The Analysis Qualifying Exam involves the tools from a) advanced calculus, b) Math 721, and c) one of the two courses: Math 722 (Complex Analysis) and Math 725 (Real Analysis). Choose one at the time of exam registration. The exam usually consists of nine questions and six are to be attempted. There will be at least two from each of a), b) and c), though some problems may involve tools from more than one area. The content of 721, 722, and 725 certainly varies somewhat from instructor to instructor. Most questions will come from the lists below. Beyond this, there may be questions from time to time involving other basic tools and techniques. Proficiency in Basic Advanced Calculus and Basic 721 and one of Basic 722, Basic 725 should suffice in order to pass the exam.

Basic Advanced Calculus
Infinite series, theorems of Bolzano-Weierstrass and Heine-Borel, uniform continuity, uniform convergence, Weierstrass approximation theorem (density of polynomials in C[a,b]), Ascoli's theorem, the Riemann integral, differentiation of series and integrals, the contraction principle, the implicit and inverse function theorems, change of variables in multiple integrals, line and surface integrals, Stokes theorem in 2 or 3 variables.
 
Basic 721
Measures, the Lebesgue integral, Lebesgue measure in Rn , notions of convergence (pointwise, almost everywhere, in measure, in mean, ... ), the monotone convergence theorem, Fatou's lemma, the Lebesgue dominated convergence theorem, Egorov's theorem, Lusin's theorem, product measures, the theorems of Tonelli and Fubini, Jensen's inequality, Lp spaces, the Riesz representation theorem, density of certain function spaces in others (including approximation by smooth functions), convolutions, differentiation and maximal function. Hilbert spaces, orthogonality, orthonormal sets, Bessel's inequality and Parseval's formula.
 
Basic 722
Cauchy-Riemann equations (both homogeneous and inhomogeneous), Cauchy's theorem, Cauchy's formula, the residue theorem, singularities, local behavior, the principle of maximum, Schwarz's lemma, analytic continuation (including the Schwarz reflection principle), Runge's theorem, theorems of Weierstrass and Mittag Leffler, normal families, conformal mapping, harmonic functions.
 
Basic 725
* Banach spaces: linear mappings, linear functionals, dual space, adjoint mapping; Hahn-Banach theorem; Baire category theorem, open mapping, closed graph, and uniform boundedness principles; duality and weak topologies, Alaoglu's theorem; basic operator theory (compact operators, perturbations of invertible operators); catlculus in Banach spaces (contraction principle and inverse function theorem); Hilbert spaces and Banach spaces of functions (Hölder spaces, Lp spaces, C(K) and its dual).
* Fourier transforms: Fourier transform and basic properties on Rd, inversion theorem, and Plancherel's theorem.
* Distributions: basic theory, Sobolev spaces, Sobolev embeddings.
 
References -- see corresponding topics in:
Ahlfors, Complex Analysis
Folland, Real Analysis
Gamelin, Complex Analysis
Royden, Real Analysis
Rudin, Principles of Math Analysis
Rudin, Real and Complex Analysis
Stein-Shakarchi, Real Analysis
Stein-Shakarchi, Complex Analysis

 

APPLIED MATHEMATICS

The Applied Mathematics Qualifying Exam consists of six problems, all of which are to be attempted. The exam is based on material usually covered in undergraduate ordinary differential equations, partial differential equations, complex variables, and the first-year graduate sequence in Applied Mathematics (Math 703-704).

ODE Theory
Existence and uniqueness for ODE; Linear systems; Solutions of equations and systems with constant coefficients; Variation of parameters; Green’s functions for ODE and solution of boundary value problems.
 
Fourier Series and Transform Method; Separation of Variables for PDE
Theory of Fourier Series; Orthogonal functions; Sturm-Liouville theory and connections with Fourier series; Special Fourier bases (Bessel functions, Legendre polynomials);Fourier transforms (Fourier and Fourier sine and Fourier cosine); Laplace transform and solution of initial-boundary value problems for equations; Evaluation of integrals via complex variables techniques.
 
Calculus of Variations
Minimization problems in finite and infinite dimension; Constrained minimization - Lagrange multipliers; Euler-Lagrange equations of an infinite dimensional variational problems (cases of systems of ODE’s and systems of PDE’s).
 
Advanced Techniques for Solutions of Partial Differential Equations
Green’s functions for elliptic, parabolic and hyperbolic problems; Conformal mapping theorem and solution of 2d Laplace equation; Method of characteristics; Self-similarity methods; Traveling waves; Dispersive waves and dispersion relations.
 
Elements of Analytical and Continuum Mechanics
Balancing laws of continuum physics; Equation of incompressible and compressible fluid mechanics; Potential theory; Modeling of spring-mass systems; Modeling of the vibrating string.
 
Asymptotic Methods
Regular pertubations; Asymptotics of integrals (Laplace method, Stationary phase).

References

Churchill, Fourier Series and Boundary Value Problems
Gelfand and Fomin B, Calculus of Variation
Kevorkian, Partial Differential Equations
Levinson and Redheffer, Complex Variables
Pinsky B, Partial Differential Equations and Boundary Value Problems
Stakgold, Green's Functions and Boundary Value Problems
Strang, Introduction to Applied Mathematics
Zanderer B, Partial Differential Equations

 

COMPUTATIONAL MATHEMATICS

The Computational Mathematics Qualifying Exam is administrated jointly between the Department of Mathematics and Department of Computer Science. Students from both departments will take the exam at the same time, but students will be given more problems than required to finish in order to fill the gap between different rules of the two departments. The problems for students from the two different departments will be slightly different.

The Mathematics Department students will have six hours to complete the exam; the material is based on Math/CS 714 and Math/CS 715. Math 714 / 715 is based on undergraduate knowledge of numerical analysis, which is covered in Math 513 / 514.

The Computer Science students will have three hours to complete the exam; the material is based on Math/CS 513, 514, 714, 717.

Covered Materials for Math Students

Numerical Methods for Ordinary Differential Equations
  • Basic ODE Theory: well–posedness
    Explicit and implicit methods, stability, Runge–Kutta and multistep methods, stiff problems
Finite Difference Methods for Parabolic Partial Differential Equations
  • Numerical differentiations, uniform and nonuniform meshes
  • Consistency, stability and convergence
    Multidimensional problems: ADI and fractional step methods
Finite Difference Methods for Hyperbolic Partial Differential Equations
  • Linear hyperbolic equations and their numerical discretizations
  • Basic theory for nonlinear hyperbolic equations: shock formation, weak solution and entropy condition, Riemann problem
  • Shock capturing methods: Godnov and Roe methods, slope limiters, flux-splitting
  • Hamilton-Jacobi equations and the level set method for front propagation
Spectral Methods for Partial Differential Equations
  • Fast Fourier transform
  • Fourier spectral method, pseudospectral methods, Chebyshev method
Numerical Algebra
  • Direct and iterative methods for linear systems, eigenvalue problems, sparse matrices, Conjugate gradient methods, nonlinear algebraic equations
Finite Element Methods For Elliptic Partial Differential Equations
  • Variational formulation, Galerkin methods, energy estimate and error analysis, implementation,
  • Discontinuous Galerkin, multigrid methods, boundary element method
Monte Carlo Methods and Molecular Dynamics
  • MC methods for integrations, random sampling, The Metropolis algorithm, molecular dynamics

References

Basic Numerical Analysis
  1. Bradie, Friendly Introduction to Numerical Analysis, Prentice Hall, 2003.
  2. Burden and Faires, Numerical Analysis, Brooks Cole, 2004. .
Finite Difference Methods
  1. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, 2007.
  2. Strikwerda, Finite Difference Schemes and Partial Differential Equations: 2nd edition, SIAM, 2004
Spectral Methods
  1. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
  2. Fornberg, Practical Guide to Pseudospectral Methods, Cambridge University Press, 1998.
Finite Volume Methods
  1. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
Finite Element Methods:
  1. Eriksson, Estep, and Hansbo, and C. Johnson, Computational Differential Equations: 2nd edition, Cambridge University Press, 1996.
  2. Zhangxin Chen, Finite Element Methods and Their Applications, Springer, 2005.
Monte Carlo Methods:
  1. Kalos and Whitlock, Monte Carlo Methods, J. Wiley & Sons, New York, 1986.

 

GEOMETRY/TOPOLOGY

To pass the qualifying exam in Geometry / Topology students must choose (at the time they register for the exam) either the algebraic topology option or the differential topology option. The algebraic topology option is based on the courses Math 751/752, and the differential topology options is based on Math 751/761.

Usually, the actual exam consists of six questions: three from basic topology and three from either algebraic topology or differential topology (as appropriate). Students are asked to write solutions to four of them. The exam is based on (a) background material usually covered in advanced calculus, undergraduate topology (e.g. 551) and undergraduate algebra courses (e.g. 541), and (b) topics from the first year graduate topology sequence (751, 752, 761), as identified below. Note that familiarity with basic concepts of point set topology (e.g. metric spaces, completeness, connectedness, and compactness) will be assumed, although these may not be treated in 751, 752, 761. The list of topics given here takes into account some changes for the  August 2014 exam.

Basic Topology:
Quotient topology, homotopy, homotopy equivalence,
retracts, deformation retracts, suspensions, joins, smash products,
fundamental group, covering spaces, cell complexes, finitely presented
groups, Seifert-Van Kampen Theorem, amalgamated products, categories,
functors, chain complexes, homology, cellular homology, the
Mayer-Vietoris Theorem, Euler characteristic, Lefschetz Fixed-Point
Theorem, calculations, and applications.
 
Algebraic Topology:
Tensor and exterior products, Hom, Ext, Universal
Coefficient Theorem, cup product, cohomology ring, Künneth Theorem,
Poincare duality and applications, Lefschetz duality, homotopy groups,
Whitehead's Theorem, cellular approximation, Hurewicz Theorem, fiber
bundles, calculations, and applications.
 
Differential Geometry and Topology:
Inverse function theorem, local coordinates, induced structures, tangent bundle, regular values, transversality, classical Lie groups, tubular neighborhoods, vector fields and flows, differential forms and de Rham cohomology, integration of forms and Stokes Theorem, relationship to singular homology, de Rham theorem, Riemannian metrics.

References

Bredon, Topology and Geometry
Guillemin and Pollack, Differential Topology
Hatcher, Algebraic Topology
Vick, Introduction to Algebraic Topology
Spivak, A Comprehensive Introduction to Differential Geometry, Volume I

 

LOGIC

The Logic Qualifying Exam will consist of a basic section plus three advanced sections, one in Model Theory, one in Computability Theory, and one in Set Theory. Students taking the exam will answer the questions in the basic section plus the questions in one of the advanced sections. Students will indicate beforehand, when they register for the logic exam, which one of the advanced sections they intend to take.

The elementary section covers material taught in 770, plus undergraduate knowledge. The advanced Model Theory, Computability Theory, and Set Theory sections correspond, roughly, to the contents of 776, 773, and 771, respectively. Thus, two logic courses (770 plus one of 776, 773, 771) should be adequate preparation for the exam.

Students should be prepared to answer questions on the following topics. Since these topics may be presented in different ways from year to year, the student should read broadly from the references to supplement the course work.

Elementary
Propositional and first-order logic syntax and semantics, Completeness and Compactness Theorems, Löwenheim–Skolem Theorem, Incompleteness Theorem, decidable and undecidable theories, axioms of ZFC, ordinal and cardinal arithmetic.

References
Ebbinghaus, Flum and Thomas, Mathematical Logic (Chs.1–6 and 10)
Shoenfield, Mathematical Logic (Chs.1–6)
Kunen, Set Theory (Chs. 1 and 3)

Computability Theory
Recursive and r.e. sets, Turing degree and jump, Recursion Theorem, strong reducibilities, arithmetic hierarchy, index sets, simple and (hyper) hypersimple sets, easy forcing arguments in recursion theory, finite and infinite injury, Friedberg-Muchnik and Sacks Splitting Theorem, Sacks Jump and Sacks Density Theorems, recursive ordinals.

References
Soare, Recursively Enumerable Sets and Degrees (Chs.1–8)
Rogers, Theory of Recursive Functions and Effective Computability (Ch.11)

Set Theory
Martin's Axiom, Suslin and Aronszajn trees, absoluteness and reflection, constructible universe, and one-step forcing constructions.

References
Kunen, Set Theory (Chs.1–7)
Jech, Set Theory (Chs. 1–4)

Model Theory
Elimination of quantifiers, types, recursive saturation, elementary chains and extensions, ultraproducts, saturated and special models, model completeness, categoricity in power, indiscernibles, o-minimal theories.

References
Chang and Keisler, Model Theory (Chs.1-3, 4.1, 4.3, 5.1, 6.1.1-2, 7.1)
Marker, Model Theory, An Introduction

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, Wi  53706

(608) 263-3054

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