Guide for Topics for the Qualifying Exams

Galois Theory

Topics:

• Field extensions including: algebraic and transcendental elements, finite/algebraic/Galois/simple/separable/purely inseparable field extensions, separable and inseparable polynomials.
• Splitting fields and algebraic closures.
• The fundamental theorem of Galois theory.
• Examples including: finite fields, polynomials of degree at most 4, composite extensions.
• Primitive elements.

Reference text
Dummitt and Foote’s Abstract Algebra book Chapter 13 (exlcuding 13.3) and Chapter 14 (excluding 14.7, 14.8, and 14.9). (110 pages).

General Algebra

You should know the meaning of and be able to give examples and non-examples of:

• Left/right/two-sided ideals, left and right modules, bimodules
• Annihilator of a module
• Matrix ring, quaternion ring, group ring
• Division ring, simple ring, zero-divisor
• Modules: Exact sequences of modules, tensor products, Hom, localization of modules, flat/projective/free modules, support of a module

References text
For a commutative ring specifically, see the references below. For a not necessarily commutative ring, see Dummitt and Foote, Chapters 7 and 10-12. (220 pages).

Commutative Algebra

Topics:

• Rings and ideals: prime/maximal/radical ideals, quotient rings, integral domains, localization of rings, local rings, polynomial rings, zero-divisors, nilpotent elements, nilradical, fraction fields, Nakayama’s Lemma.
• Modules: see the list in “general algebra.”
• Noetherian rings, including chain conditions and the Hilbert Basis Theorem.

Reference text
In recent years 742, which concentrates on commutative algebra, has been taught from Altman-Kleiman’s A term of commutative algebra, and the relevant chapters would be 1-5 and 8-13. (55 pages).

An alternate source would be Atiyah-MacDonald’s Introduction to Commutative Algebra chapters 1-3 and 6-7. (88 pages).

Group Theory

You should know the meaning of, and be able to give an example and a non-example of the following:

• Group
• order (of a group)
• order (of a group element)
• normal subgroup
• quotient group
• abelian group, nilpotent group, lower central series, solvable group, simple group, perfect group
• commutator subgroup, centralizer, normalizer, conjugacy class
• group homomorphism
• group action, orbit, stabilizer, transitive action, faithful action
• free group, finitely presented group
• p-group, symmetric group, permutation group, alternating group, dihedral group, general linear group

You should be able to:

• State and apply the orbit-stabilizer theorem;
• Compute the conjugacy classes of a finite group;
• Work fluently with free groups, matrix groups, and symmetric groups

Reference text
Dummit and Foote Chapters 1, 2.1-2.4, 3.1-3.3, 4.1-4.3, 5.1-5.2, 5.4, 6.1, and 6.3.

Linear Algebra

You should know the meaning of, and be able to give an example and a non-example of the following:

• Eigenvalue, eigenvector, generalized eigenspace
• Jordan normal form
• dual vector space, transpose, bilinear form, Hermitian form
• orthogonal matrix, symplectic matrix
• tensor product of vector spaces

References text
Dummit and Foote, chapters 11-12.

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 Complex Analysis, commensurate with the Math 722 course at UW Madison.

Recommended texts
Function theory of one complex variable by Greene and Krantz; Functions of one complex variable by J.B. Conway.

Good sources for additional reading and problems: old qualifying exams, Gamelin’s Complex Analysis, Rudin’s Real and Complex Analysis, Stein-Shakarchi: Princeton Lectures in Analysis II: Complex Analysis.

List of topics:
• Analytic functions and Cauchy-Riemann equations. Elementary functions, branches and principal branches.
• Line integrals. Cauchy’s theorem and Cauchy’s formula.
• Cauchy’s estimates, Liouville’s theorem, Morera’s theorem, Goursat’s theorem.
• Power series, Laurent series, and isolated singularity. Residue calculus.
• Argument principle, Rouche’s theorem, Hurwitz’s theorem, open mapping theorem.
• Simply connected domains. Normal families and Montel’s theorem.
• Conformal mappings of the unit disc and upper half-plane, fractional linear transformations. Schwarz’s lemma. Elementary conformal mappings. The Riemann Mapping Theorem.
• Harmonic functions, the mean value property and maximum principle, Harnak’s lemma and principle, subharmonic functions.
• Dirichlet problem on the unit disc. Schwarz reflection principle. Perron’s theorem.
• Mittag-Leffler’s theorem, Runge’s 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 III: Real 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:   Hilbert space theory.
  

  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 text
  Chapter 6 of Folland.

• 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 text
  Chapter 5 of Folland.

• Fourier series and transforms.
  

  Chapter 8 of Folland, Chapter 7 in Rudin.

• Distributions.
  

  Chapters  1,  6, 7  Rudin.

• Sobolev spaces.
  

  9.3 of Folland.

• Further consequences of the Baire Category Theorem.
  

  Chapter 4 of Stein and Shakarchi (omitted in 2021)

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),WKB, boundary layer analysis and the Method of Multiple Scales.

References

Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers
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

The Computational Mathematics Qualifying Exam is offered at the beginning of every fall and spring semester. Students have 6 hours to complete the exam of about 5-6 problems. The exam is typically 120 points in total. The material is based on Math/CS 714 and Math/CS 715. The students taking Math 714 / 715 are assumed to have a basic understanding of the undergraduate level of numerical analysis (covered in Math 513 / 514).

Covered Materials for Math Students

All materials that may appear in the exam are listed below. The components that are marked “advanced” are unlikely to appear, but we do not rule out the possibility. (Please contact the most recent instructor of 714/715 for details.)

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

• Numerical differentiations, uniform and nonuniform meshes
• Consistency, stability and convergence
• Multidimensional problems: ADI and fractional step methods

Finite Volume 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
• (advanced) Shock capturing methods: Godnov and Roe methods, slope limiters, flux-splitting
• (advanced) 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
• (advanced) Discontinuous Galerkin, multigrid methods, boundary element method

References

Many textbooks cover similar topics. If textbook (A) is listed under topic (b), that means we believe (A) organizes materials (b) better than other textbooks. However, every student is different. Ultimately please choose textbooks according to your own preferences. We only list recommendations below.

Basics:

Basic Numerical Analysis

1. Suli and Mayer, An Introduction to Numerical Analysis

Finite Difference Methods

1. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, 2007.

Spectral Methods

1. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
2. Gottlieb and Orzag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, 1977.

Finite Element Methods:

1. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009.
2. Larson and Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013.

Advanced:

Finite Volume Methods

1. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.

Monte Carlo Methods:

1. Kalos and Whitlock, Monte Carlo Methods, J. Wiley & Sons, New York, 1986.

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 text for 761 is John Lee’s Introduction to Smooth Manifolds.
Ch 1-6
8 up to Lie brackets
9 up to Lie derivatives
12,14
15 up to Orientations of Manifold
16 up to Stoke’s Theorem
17 up to Homotopy Invariance

Additional reference texts for 761 are 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 covered 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, orientation, integration of forms, distributions, basic 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.

The Logic Qualifying Exam will consist of (usually 6) questions based on the content of the two introductory graduate courses: 770 and 773.