Qualifying Exams typically take place the week or two before classes begin each semester (except summer); a precise schedule is posted months in advance. 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.  You can get previous years' exams in the Math Library.

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).  A thorough knowledge of most of the items below should be sufficient to pass.

I.       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.

II.      Ring Theory:

a)            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.

b)            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.

III.     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.

IV.     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                                    

Isaacs -- Algebra: A Graduate Course (skip Ch.15 & 25)

Hoffman and Kunze -- Linear Algebra                                               

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 and Math 725 (your choice at the time of exam sign-up). The exam consists of 9 questions and 6 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. However, a good part of it does not, and 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.

I. Basic Advanced Calculus
 

Infinite series, theorems of Bolzano-Weierstrass and Heine-Borel, uniform continuity, uniform convergence, Weierstrass 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.

II. Basic 721
 

Measures, the Lebesgue integral, Lebesgue measure in /R^n /, 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, L^p 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.

III. 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.

IV. 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); calculus in Banach spaces (contraction principle and inverse function theorem); Hilbert spaces and Banach spaces of functions (Hoelder spaces, L^p spaces, C(K) and its dual).  

*Fourier transforms: Fourier transform and basic properties on R^d, 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/
Rudin --/ Functional Analysis/
Stein -- /Real analysis/
Stein --  /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).

(v)        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.

(vi)       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.

(vii)      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).

(viii)      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.

(ix)       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.

(x)        Asymptotic Methods

Regular pertubations; Asymptotics of integrals (Laplace method, Stationary phase).

References:

Churchill -- Fourier Series and Boundary Value Problems

Gelford and Formin 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

LOGIC

The Logic Qualifying Exam will consist of a basic section plus three advanced sections, one in Model Theory, one in Recursion 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 sign up for the logic exam, which of the advanced sections they intend to take.

The elementary section covers material taught in 770, plus undergraduate knowledge.  The advanced Model Theory, Recursion 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.

I.    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)

II.   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)

III.  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), or

Jech,  Set Theory (Chs. 1-4)

IV.  Model Theory:

Elimination of quantifiers, types, recursive saturation, elementary chains and extensions, ultraproducts, saturated and special models, model completeness, categoricity in power, indiscernibles.

References:

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

Hodges B A Shorter Model Theory

  TOPOLOGY

The Topology Qualifying Exam consists of six problems: three in general and algebraic topology, and three in differential topology.  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.

I.          Basic topological notions: quotient topology and examples such as suspensions, joins, smash products.  Compactly generated topologies, neighborhood retracts.  Cell complexes.

II.         Differentiable manifolds: inverse function theorem, local coordinates, induced structures.  Tangent vectors, tangent bundle.  Regular values, transversality.  Classical Lie Groups.  Tubular neighborhoods, vector fields and flows.

III.        Homotopy, fundamental group, covering spaces, amalgamated products, finitely presented groups, Seifert-Van Kampen Theorem, elements of knot theory, calculations and applications.

IV.        Homological algebra:  categories and functors, chain complexes, tensor and exterior products, Hom, Ext, Universal Coefficient Theorem, Künneth Theorem.

V.         Singular and cellular homology.  Basics such as the Eilenberg‑Steenrod Axioms and the Mayer‑Vietoris Theorem. Cellular chain complexes and the homology of CW complexes. Computations of the homology of simple spaces such as spheres, surfaces, projective spaces, lens spaces, etc. Applications to nonretraction theorems, Brouwer fixed-point theorem, separation theorems, the fundamental theorem of algebra.

VI.        Differential forms and de Rham cohomology.  Integration of forms and Stokes Theorem.  Relationship to singular homology, de Rham theorem.

VII.       The cohomology of a space, cup and cap products.  Duality Theorems such as Poincaré duality, Lefschetz duality and applications.  Euler class and the Lefschetz Fixed-Point Theorem.

References:

Bredon, Topology and Geometry

Munkres, Topology

Greenberg and Harper, Algebraic Topology: A First Course

Guillemin and Pollack, Differential Topology

Hirsch, Differential Topology

Massey, Algebraic Topology: An Introduction

Munkres, Elements of Algebraic Topology

Vick, Introduction to Algebraic Topology        

 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 materials are based on Math/CS 714 and Math/CS 715.

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

Covered Materials for Math Students 

I           Numerical Methods for Ordinary Differential Equations

  ‑ Basic ODE Theory: well‑posedness

-          Explicit and implicit methods, stability, Runge‑Kutta and multistep methods, stiff problems

II.          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 

III.         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

IV.        Spectral Methods for Partial Differential Equations

  ‑ Fast Fourier transform

  ‑ Fourier spectral method, pseudospectral methods, Chebyshev method

V.         Numerical Algebra:

  ‑ Direct and iterative methods for linear systems, eigenvalue problems, sparse matrices,

    Conjugate gradient methods,  nonlinear algebraic equations

VI.        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

VII.       Monte Carlo Methods and Molecular Dynamics

  ‑  MC methods for integrations, random sampling, The Metropolis algorithm,  molecular

     dynamics

References

‑‑‑‑‑‑‑‑‑‑

J. Strikverda: Finite Difference Schemes and Partial Differential Equations , Wadsworth &

Brooks/Cole Advanced Books & Software, 1989

R.J. LeVeque: Numerical Methods for Conservation Laws, Birkhauser, 1992

D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods, SIAM 1977 J. Stoer and

 R. Bulirsch, Introduction to Numerical Analysis, second edition, Springer‑Verlag, 1993.

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method,

Cambridge University Press, 1987

M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, J. Wiley & Sons, New York, 1986.

Study strategies 

 Previous qualifying exam survivors have identified these invaluable strategies to help you prepare for the exams:

(i)         Access old exams and most solutions at http://math.library.wisc.edu/quals.htm.  You can also check out a copy of the old exams from the library.  These can give you a good idea of the types of questions you’re likely to see, help you review the content, and get a feel for the structure of the exams.  There are multiple copies of the exam books in the library.  Be sure to check through them carefully, since some of them are missing an exam or two (or their solutions).

 (ii)       Form a study group with other students who are preparing for the same exam.  By working with other students on old exams, homework problems, and general concepts, you will strengthen your understanding and preparation for the exam.

(iii)       Talk with graduate students who have passed the exam in that area.  They can provide you with valuable hints and insights, and might even offer to answer questions you have as you study.

(iv)      Ask faculty questions.  While there are fewer people around the department in the summer and over winter break, there are professors around, and you may find many of them helpful as you’re studying.  Don’t hesitate to ask!  You have nothing to lose.

Exam Tips  In addition, here are some specific tips for each of the exams.  While some of these “facts” are subject to change, they do reflect patterns in the exams over the past few years, so use them as guidelines.

Algebra  The exam is generally five questions, with at least one problem in each of groups, rings, linear algebra, and Galois theory (the fifth question is a combination of these four). Groups, rings, and galois theory are covered in the first year courses (741-742), but linear algebra is not.  If you need more work in linear algebra, consider taking Math 542, which will help fill in the necessary background.

Analysis  The exam is based on 721 (real analysis) and your choice of 722 (complex) or 725 (functional analysis).  It usually contains at least a couple of problems which have appeared on previous exams.  However, be aware that the format of the exam changed around 1997, so exams older than that will be less helpful.

Applied Mathematics  The number of questions varies from exam to exam, but you usually will have some choice of questions to answer--be sure to read the directions carefully.  A good background in complex analysis can be a big help; if your background in this area is weak, you will need to do some extra reading, or consider taking Math 623. 

Computational Mathematics  The exam is based on Math/CS 712 and 713.  There are usually five to six problems.  Previous exams could be of help in knowing the type of problems being tested.

Logic  The elementary section of the exam is based on 770 (foundations) and the second section is based on one of 771 (set theory), 773 (recursion theory), and 776 (model theory).  Each section contains three questions, of which you must choose two.  The exam is generally quite consistent, but as with all quals, varies slightly depending on who taught the first-year courses.

Topology  The exam usually contains 6 or 7 questions, of which you can select four to answer.  It’s common for exam questions to come from the previous semesters’ homework problems, as well as recent exams.  There are generally questions on general topology, covering space theory and the fundamental group, homology, and cohomology.  The first year course in topology (751-752) includes a lot of algebra, much of which is developed as you need it.