• Algebra: Properties of rings: Fundamental aspects of ring and module theory are covered, Euclidean rings, Principal Ideal Domains, Unique Factorization Domains, fields, field extensions, finite extensions, algebraic extensions, algebraically closed fields, Descending chain condition and Artinian rings, polynomial rings, matrix rings, ascending chain condition and Noetherian rings, finitely generated modules, direct sums of modules, free modules, invariant basis number. Properties of groups: Elementary theory of groups, automorphisms, split extensions, Sylow theorems, examples including dihedral groups, quaternion groups and other groups of small order, p-groups, nilpotent groups, solvable groups and simple properties of such groups, abelian groups, quasi-cyclic groups, finitely generated groups, free groups and their construction, wreath products, groups with the maximum condition or the minimum condition, simple groups.
  • The Algebra Qualifying Exam is based on Math 571 and Math 573. The material covered in Math 571 may vary depending upon the interests of the professor who teaches the course. Part II of Dummit and Foote will be covered in Math 571 together with additional topics from Parts III, IV and V. The Qualifying Exam will have enough problems on it to satisfy the needs of all students, irrespective of who taught the course. The material covered on the Algebra Qualifying Exam can be found in the books of D. S. Dummit and R. M. Foote, Abstract Algebra, and J. J. Rotman, An Introduction to the Theory of Groups. The algebra exam usually includes definitions, statements and proofs of theorems, examples, and standard exercises.
• Analysis: Sigma algebras and Lebesgue measure, measurable functions; Lebesgue integration; monotone convergence theorem, Fatou’s lemma, dominated convergence theorem; Product measures, Tonelli’s theorem and Fubini’s theorem; Abstract measures, signed measures, Jordan decomposition theorem, Radon-Nikodym theorem; Lp spaces, Holder’s and Minkowski’s inequality, dual spaces; Differentiation theory, bounded variation, absolute continuity, Lebesgue differentiation theorem; Hilbert spaces, bounded operators and their adjoints; Elementary properties of Banach spaces, dual spaces, Hahn- Banach theorem, open mapping theorem, closed graph theorem, uniform boundedness property.
  • The courses preparatory to the analysis exam are Math 580 and Math 681. Most of the above material can be found in Royden’s book, Real Analysis. Students should be familiar with a substantial collection of examples and counterexamples, and with the proofs of standard theorems.
• Numerical Analysis: The material covered in the qualifying exam in numerical analysis is based on the core courses Math 511-Math MA512, and includes: Error analysis, solution of linear and nonlinear systems of equations, eigenvalues, interpolation and approximation, least squares problems, numerical differentiation, integration and Richardson extrapolation, initial and boundary value problems for ordinary differential equations, finite difference methods for solving partial differential equations, stability analysis of numerical schemes, basic iterative methods needed for solving elliptic equations, and finite element methods for solving elliptic problems. Background material on linear algebra is assumed: This includes Gaussian elimination and matrix factorization, vector spaces, linear dependence and independence and bases.
  • The book Numerical Analysis by Richard L. Burden and J. Douglas Faires, Brooks-Cole, Cengage Learning, August 2010, (chapters 2-8, and chapters 10-12) is often used in Math 511-Math 512. It covers in sufficient detail all the material listed above.
• Partial Differential Equations: The first part of the exam is concerned with solutions for the heat, wave, and Laplace’s equations in bounded domains. Topics include the L2 theory of Fourier series, the formal differentiation and integration of Fourier series, the eigenvalues and eigenfunctions for elliptic operators (specifically the Laplace operator) with Dirichlet and Neumann boundary conditions, orthogonality of eigenfunctions, orthogonal expansions, solutions of both homogeneous and nonhomogeneous boundary value problems in Cartesian and polar coordinates, existence and uniqueness of solutions for the heat, wave, and Laplace’s and Poisson’s equations, Fredholm alternative. The pertinent topics for the second part of the exam are: First-order equations (The Cauchy problem for quasilinear equations; Method of characteristics; Semi-linear equations; Weak solutions; Conservation laws, jump conditions, fans and rarefaction waves; General nonlinear equations); Second-order equations (Classification by characteristics; Canonical forms and general solutions; First-order systems; Well-posedness and Cauchy problems; Cauchy-Kovalevski theorem; Adjoints and weak solutions; Transmission conditions, delta distributions, convolution and fundamental solutions); The wave equation (Initial value problems; Weak solutions; Duhamel’s principle; Spherical means; Hadamard method of descent; Domains of dependence and influence; Huygen’s principle; Energy methods; Traveling wave solutions); Laplace equation (Green’s formulas; Separation of variables; Spherical Laplacian in R3 and Rn for radial functions; Mean value theorem; Maximum principle; The fundamental solution; Green’s functions and the Poisson kernel; Properties of Harmonic functions; Eigenvalues of the Laplacian; Method of eigenfunctions expansion; Dirichlet problem on a half-space; Dirichlet problem on a ball; Helmholtz decomposition); Heat equation (The pure initial value problem; Fourier transforms; Non-homogeneous equations; Similarity solutions; Regularity; Energy methods; Maximum principle, uniqueness and fundamental solution).
  • Currently, the two recommended books used in MATH 541 and MATH 642 are Partial Differential Equations: An Introduction, 2nd ed., by Walter Strauss and Partial Differential Equations, 2nd ed., by Lawrence C. Evans.
• Topology: Topological spaces, metric spaces, Baire Category Theorem, separation and countability axioms, compactness and related concepts, connectedness and related concepts, continuous functions, Urysohn’s Lemma, Tietze’s Extension Theorem, spaces of functions, Tychonoff’s Theorem, quotient spaces, CW-complexes, homotopy of continuous functions, fundamental group, covering spaces and lifting criteria, singular homology, Hurewitz Theorem, exact sequences, Euler Characteristic, and computations of certain fundamental and homology groups.
  • Courses preparatory to the topology exam are Math 565 and Math 566. Most of the material can be found in the book of Fred H. Croom, Principles of Topology or of James R. Munkres, Topology First Course; the first 21 sections in Greenberg-Harper, Algebraic Topology First Course, or Chapters 1 and 2 in Hatcher, Algebraic Topology. Normally, the relevant two-course sequences are designed to help students prepare for the Qualifying Exam. However, students are responsible for all listed topics although some topics may not be covered in class because of time constraints.

Option 2: Pass a comprehensive exam that is based on the plan of study.

Instead of taking two written qualifying exams, students may choose to take a comprehensive exam led by their dissertation advisors. For details see the graduate catalog.

The Joint Program Exam for PhD Students in Applied Mathematics

• Linear and Numerical Linear Algebra: Vector spaces over a field. Subspaces. Quotient spaces. Complementary subspaces. Bases as maximal linearly independent subsets. Finite dimensional vector spaces. Linear transformations. Null spaces. Ranges. Invariant subspaces. Vector space isomorphisms. Matrix of a linear transformation. Rank and nullity of linear transformations and matrices. Change of basis. Equivalence and similarity of matrices. Dual spaces and bases. Diagonalization of linear operators and matrices. Cayley-Hamilton theorem and minimal polynomials. Jordan canonical forms. Real and complex normed and inner product spaces. Cauchy- Schwartz and triangle inequalities. Orthogonal complements. Orthonormal sets. Fourier coefficients and the Bessel inequality. Adjoint of a linear operator. Positive definite operators and matrices. Unitary diagonalization of normal operators and matrices. Orthogonal diagonalization of real symmetric matrices. Bilinear and quadratic forms over a field. Triangular matrices and systems. Gaussian elimination. Triangular decomposition. The solution of linear systems. The effects of rounding error. Norms and limits. Matrix norms. Inverses of perturbed matrices. The accuracy of solutions of linear systems. Orthogonality. The linear least squares problem. Orthogonal triangularization. The iterative refinement of least squares solutions. The space Cn. Eigenvalues and eigenvectors. Reduction of matrices by similarity transformations. The sensitivity of eigenvalues and eigenvectors. Hermitian matrices. The singular value decomposition. Reduction to Hessenberg and tridiagonal forms. The power and inverse power methods. The explicitly shifted QR algorithm. The implicitly shifted QR algorithm. Computing singular values and vectors. The generalized eigenvalue problem.
• Mathematical Analysis: (1) sup and inf for subsets of R, limsup, liminf for real sequences, Bolzano-Weierstrass theorem, Cauchy sequences. (2) Continuous functions: min-max, intermediate value theorem, uniform continuity, monotone functions. (3) Derivative: mean value theorem, Taylor’s theorem for real functions on an interval. (4) Riemann integration for functions on an interval. Improper integrals. Integrals depending on parameters. (5) Sequences of functions: pointwise and uniform convergence, interchange of limits. (6) Series of functions: M-test, differentiation/integration, real analytic functions. (7) Metric spaces: open and closed sets, completeness and compactness, Cauchy sequences, continuous functions between metric spaces, uniform continuity, Heine-Borel and related theorems, contraction mapping theorem, Arzela-Ascoli theorem.