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 (Exam Syllabus for 2018-2019)

 

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. Questions for 2018-2019 will come from the topics and tools below.

Basic Advanced Calculus
Basic undergraduate analysis, commensurate with the Math 521-522 sequence at UW Madison.
Study guide: These courses are typically taught at the level of Rudin's Principles of Mathematical Analysis.
A syllabus for this sequence may be found here.
This list of practice problems consisting mainly of demanding undergraduate analysis problems may be a  good start for those looking to review this material.
 
Complex Analysis
Graduate Real Analysis, commensurate with the Math 722 course at UW Madison.
Recommended texts: The principal reference is Gamelin's Complex Analysis, chapters 1-14.
Good sources for additional problems:  old qualifying exams,  Rudin’s Real and Complex Analysis, Stein-Shakarchi: Princeton Lectures in Analysis II: Complex Analysis.

List of topics:  

  • Continuous branches of multi-valued functions, principal branches of elementary functions. Analytic functions and Cauchy-Riemann equations.
  • Green's theorem. Cauchy's theorem and Cauchy's formula. Harmonic functions and mean-value property. Maximum principle.
  • Line integrals and path independence of line integrals. Harmonic conjugate. Analytic continuation.
  • Liouville's theorem, Morera's theorem, Goursat's theorem. Pompeiu's formula.
  • Power series, Laurent series, and isolated singularity. Residue calculus. 
  • Argument principle, Rouche's theorem, Hurwitz's theorem, open mapping theorem. Winding numbers and Jump theorem for Cauchy integrals.
  • Conformal mappings of the unit disc and fractional linear transformations. Schwarz' lemma, Pick Lemma. Dirichlet problem on the unit disc.  Schwarz reflection principle. 
  • Simply connected domains. Normal family and the proof of Riemann mapping theorem. Schwarz-Christoffel formula
  • Marty's theorem, Montel's theorem, and Picard theorems. 
  • Runge's theorem, theorems by Weierstrass and Mittag-Leffler. 
  • The Gamma function, Laplace Transforms, the Zeta function, Dirichlet Series, and the prime number theorem. 
 
Measure, Integration and the Fundamentals of Functional Analysis
First semester graduate Real Analysis, commensurate with the Math 721 course at UW Madison.
Recommended texts:
The principal reference is Folland's Real Analysis:  Modern Techniques and Their Applications, Chapters 1-5.
 
Good sources for additional problems:  old qualifying exams,  Rudin’s Real and Complex Analysis, Chapters 1-8.  Chapter 2 of Rudin’s Functional Analysis (for problems on the Baire Category Theorem). Stein-Shakarchi: Princeton Lectures in Analysis IIIReal Analysis.

List of topics:  

  • Measure and integration, with a particular emphasis on: measures, the Lebesgue integral and measure on R^n, modes of convergence (pointwise, almost everywhere, in measure, in mean), the Monotone Convergence Theorem, Fatou’s Lemma, the Dominated Convergence Theorem, Egorov’s Theorem, Lusin’s Theorem, product measures, the Fubini and Tonelli Theorems.  Reference:  Chapters 1 and 2 of Folland
  • Signed measure and differentiation, with a particular emphasis on the Radon-Nikodym Theorem, the Lebesgue Differentiation Theorem in R^n, and the connection between differentiation theorems and bounds for maximal functions.  Reference: Chapter 3 of Folland, excluding functions of bounded variation.
  • Basic point set topology, commensurate with Chapter 4 of Folland, particularly non-metric topologies, locally compact and locally convex spaces.
  • Introductory functional analysis:  Banach spaces, linear mappings, linear functionals, duality, adjoint mapping, Hahn-Banach Theorem, Baire Category Theorem, Open Mapping and Closed Graph Theorems, Principle of Uniform Boundedness, the weak and weak-* topologies, operator topologies (norm, strong, weak), perturbations of invertible operators, Hilbert spaces.  Reference: Chapter 5 of Folland.

 

Functional analysis, distributions and the Fourier transform.
            
Second semester Real Analysis, commensurate with Math 725 at UW Madison.
Recommended texts. Details are given in the list of topics.

  • Folland, Chapters 6-9.
  • Rudin’s Functional Analysis, Chapters 6-8. (Distribution theory is typically taught at the level of Rudin’s Functional Analysis, rather than Folland.
  • Stein and Shakarchi’s Princeton Lectures in Analysis IV: Functional Analysis, Chapter 4. (Good reference and problems for further consequences of the Baire Category Theorem.) 
List of topics:

 

  • Lp spaces and their duals.  Reference: Chapter 6 of Folland.
  • Radon measures and the Riesz Representation Theorem:  Chapter 7 of Folland.
  • Fourier series and transforms:  Chapter 8 of Folland.  
  • Distributions:  Chapter 6-7  Rudin.
  • Sobolev spaces:  9.3 of Folland.
  • Further consequences of the Baire Category Theorem:  Chapter 4 of Stein and Shakarchi

 

 

           

  

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 (Exam Syllabus 2018-2019) 

Logistics of the exam:

When registering for the exam, students must choose 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 option is based on Math 751/761.

The exam consists of two parts, Part I and Part II. Each part has three questions. Part I is the same on both exams, and covers material from 751.  Part II of the Algebraic Topology option covers material from 752, and Part II of the Differential Topology option covers material from 761.

Students are asked to answer two questions from Part I and two questions from Part II.

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.

Reference texts:

The reference text for 751 and 752 is Allen Hatcher’s Algebraic Topology.

The reference texts for 761 are John Lee’s Introduction to Smooth Manifolds, Frank Warner’s Foundations of Differentiable Manifolds and Lie Groups, and Spivak’s A Comprehensive Introduction to Differential Geometry, Volume I.

Description of advanced material coverered by the exam.

Part I. The student should be prepared to:

  • Work with the standard constructions in algebraic topology, such as homotopies, chain complexes, quotients, products, suspensions, retracts, and deformation retracts.
  • Effectively use the fundamental tools of homology, reduced homology, the long exact sequence of a pair, excision, and the Mayer-Vietoris sequence for homology.
  • Compute the fundamental group of an explicitly given cell complex.
  • Compute the fundamental group of a space using the Seifert-Van Kampen Theorem.
  • Compute the homology of an explicitly given cell complex using the definition of cellular homology.
  • Make use of the standard cell structures of spheres and real and complex projective spaces in all dimensions.
  • Know the fundamental and homology groups of spheres and real and complex projective spaces in all dimensions.
  • Compute the homology of a space using the Mayer-Vietoris sequence.
  • Make use of the long exact sequence in homology to make computations.
  • Compute the Euler characteristic of a space.
  • Construct finite covering spaces of an explicitly given cell complex.
  • Construct covering spaces with prescribed group of deck transformations by constructing a corresponding quotient of the fundamental group.
  • Use contractibility of the universal cover to deduce that certain maps are null-homotopic.
  • Use local homology to distinguish two spaces.
  • Use the Lefschetz fixed point theorem to find a fixed point of a continuous map.
  • Combine the above machinery and techniques to solve problems.

 

Part II.  Algebraic option. The student should be prepared to:

  • Compute the cohomology of an explicitly given cell complex using the definition of cellular cohomology.
  • Effectively use the fundamental tools of cohomology, reduced cohomology, cup product, cap product, cross product, the long exact sequence of a pair, excision, and the Mayer-Vietoris sequence.
  • Apply the Universal Coefficient Theorem in computations.
  • Compute cup products of cohomology classes.
  • Distinguish the homotopy types of two spaces using the Cohomology Ring.
  • Make effective use of Poincaré duality.
  • Make elementary computations of homotopy groups using the Hurewicz Theorem.
  • Know the homotopy groups of the n-sphere through dimension n.
  • Know the homotopy groups of the 2-sphere through dimension 3.
  • Build continuous maps between cell complexes inductively using high-connectivity of the target: e.g. “Using the fact that Y is k-connected, construct a map from the given X to Y.”
  • Make effective use of Whitehead’s Theorem.
  • Recognize and construct fiber bundles.
  • Use the long exact sequence of homotopy groups of a fibration.
  • Know the standard examples of fiber bundles of spheres over spheres arising from the unit spheres in the real division algebras.
  • Combine the above machinery and techniques to solve problems.

 

Part II. Differential Option. The student should be prepared to:

  • Work with the standard concepts in differential topology, including smooth manifolds, local coordinates, transversality, regular values, the Inverse Function Theorem, tubular neighborhoods, vector fields, flows, differential forms, integration of forms, distributions, de Rham cohomology, and Stokes Theorem.
  • Perform computations with differential forms, including integration of explicit forms over given submanifolds.
  • Perform computations with the Lie derivative.
  • Make use of Sard’s theorem.
  • Distinguish de Rham cohomology classes given explicit forms on an explicit manifold.
  • Show that a given manifold admits a smooth structure.  For example, the student should be able to show that spheres, projective spaces, Grassmannians, the special linear group, and the orthogonal group admit smooth structures.
  • Construct trivializations of explicit vector bundles, such as the tangent bundle of the 3-sphere.
  • Combine the above machinery and techniques to solve problems.

 

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

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