Doctoral students must take quals in three of the six available
areas: (1) Numerical Analysis/Linear Algebra, (2) Analysis, (3) Algebra,
(4) PDEs, (5) Algebraic Topology, and (6) Geometry. Please see the
for more details on the exams.
Register for the next round of qualifying exams >
Examination Dates and Registration: Click below to see dates for
upcoming exams and to signup. Students must sign up at least 2 weeks
prior to the date of the exam, and may withdraw before that 2 week
deadline. Any student who is still signed up by the 2-week mark and does
not show for the exam, will receive a No Pass.
Core Examination Topics: These are the specific
topics that may be covered on a given exam.
(1) Numerical Analysis/Linear Algebra
Systems of Equations
Linear Systems of Equations:
Gaussian elimination, LU- and Cholesky decompositions
for full and sparse matrices, operation counts,
stability of linear systems (condition number),
stability of Gaussian elimination. Basic iterative
methods (Jacobi, Gauss-Seidel, Successive Overrelexation
method), Conjugate Gradient Method
Gerschgorin theorem, power method, inverse power method,
stability of eigenvalue problems
Nonlinear Systems of Equations, Optimization:
Newton's method, quasi-newton methods, fixed point
iteration. Newton and Levenberg-Marquardt methods for
Lagrange and Hermite interpolating polynomials, Runge
phenomena. Splines, least squares approximation of
functions and orthogonal polynomials.
Newton-Cotes methods, Gaussian quadrature, Euler-MacLaurin
formula, Adaptive quadrature.
Convergence of explicit one-step methods, Stiffness, A-
stability, impossibility of A-stable explicit
- An Introduction to Numerical Analysis by K. E.
- Unconstrained Optimization by P. E. Frandsen, K.
Jonasson, H. B. Nielsen, and O. Tingleff
- Analysis of Numerical Methods by E. Isaacson and H. B.
Keller (Wiley, Dover reprint)
- Finite Difference Methods for Ordinary and Partial
Differential Equations by R. LeVeque (SIAM)
- A First Course in the Numerical Analysis of
Differential Equations by A. Iserles (Cambridge
In addition to the topics covered below, students are
expected to be proficient in the topics covered in an
undergraduate sequence in Real and Complex Analysis. These are
the topics covered in our Math 133, 135, and 136 courses at
Metric Spaces: Stone-Weierstrass theorem,
Arzela-Ascoli theorem, Baire category theorem.
Measure and Integration: General measure spaces
(including sigma-algebras, measures, completions), Caratheodory
Extension theorem, Borel sets, Lebesgue measure. Cantor set and
Cantor function. Measurable functions, abstract integration,
Lebesgue integral and limit theorems for integrals. Complex
measures, total variation measure, absolutely continuous,
singular measures, and Radon-Nikodym theorem. Product measures
and Fubini-Tonelli theorem.
Functional Analysis: Banach spaces, Lp spaces,
the Holder and Minkowski inequalities, the Banach-Steinhaus,
Open Mapping, and Hahn-Banach theorems. Hilbert spaces,
orthogonal decompositions and projections, orthonormal bases,
Fourier series. Dual spaces, operator norms, adjoints, duality
of Lp and Lq. Riesz representation theorem.
Complex Analysis: Analyticity, Cauchy-Riemann
equations, elementary functions, Cauchy's theorem, Cauchy
integral formula, Liouville's theorem, Morera's theorem, Taylor
expansions, classification of singularities, Laurent expansions,
the residue theorem, linear fractional transformations.
- Real Analysis by H. L. Royden
- Real and Complex Analysis by Walter
- Real Analysis: Measure Theory,
Integration, and Hilbert Spaces by E. M.
Stein and R. Shakarchi (Princeton University
- Complex Analysis by E. M. Stein and
R. Shakarchi (Princeton University Press)
- Functional Analysis: Introduction to
Further Topics in Analysis by E. M. Stein
and R. Shakarchi (Princeton University Press)
- Fourier Analysis: An Introduction
by E. M. Stein and R. Shakarchi (Princeton
- Quotients and Isomorphism Theorems for groups, rings, and
- The action of a group on a set; applications to conjugacy
classes and the class equation.
- The Sylow theorems; simple groups.
- Simplicity of the Alternating Group for n≥5n≥5.
Rings and Modules:
- Polynomial rings, Euclidean domains, principal ideal domains.
- Unique factorization; the Gauss lemma and Eisenstein's
criteria for irreducibility.
- Free modules; the tensor product.
- Structure of finitely generated modules over a PID;
applications (finitely generated abelian groups, canonical forms
of linear transformations).
- Algebraic, transcendental, separable, and Galois extensions,
- Finite fields, algebraic closures.
- The fundamental theorem of Galois theory for a finite
extension of a field of arbitrary characteristic.
- Basic Algebra I by Nathan Jacobson (W. H. Freeman)
- Algebra by Thomas W. Hungerford (Springer)
- Algebra (in part) by Serge Lang (Addison-Wesley)
- Algebra by Michael Artin (Prentice Hall)
- Abstract Algebra by David S. Dummit and Richard M. Foote
(4) Partial Differential Equations
Linear Partial Differential Equations
Laplace, Poisson and Helmholtz equations, boundary-value
problems, existence and uniqueness, Fredholm alternative, weak
and strong elliptic maximum principles, boundary regularity,
Sobolev spaces, weak solutions, Lax-Milgram Theorem, Galerkin
method, variational principles, Fréchet derivative.
Heat equation, Schrödinger equation, existence and uniqueness of
solutions, weak and strong parabolic maximum principles,
Wave equation, method of characteristics.
Fourier series, Fourier transforms, convergence and
approximation properties, generalized functions, distributions.
Eigenfunction expansion, Sturm-Liouville Theory, Rayleigh
quotient, Rayleigh-Ritz method, Green's functions.
Quasilinear and Nonlinear PDE:
Cauchy-Kovalevskaya theorem, hyperbolic systems, shallow-water
equations, gas-dynamic equations, Fourier methods, energy
methods. Method of characteristics, weak solutions, jump
conditions, entropy conditions.
- Basic Linear Partial Differential Equations by F. Treves
(Academic Press, Dover reprint)
- An Introduction to Nonlinear Partial Differential Equations by
J. D. Logan (Wiley)
- An Introduction to Partial Differential Equations by Pinchover
- Applied Function Analysis by Griffel
- Partial Differential Equations by L. C. Evans (AMS, Second
(5) Algebraic Topology
Homotopy and Fundamental Group:
- Homotopy and homotopy equivalence.
- Fundamental group - including key examples.
- Van Kampen's Theorem - basic calculations
using cell complexes.
- Basic lifting properties.
- Universal covering.
- Relation between coverings and subgroups of
the fundamental group.
- Group actions.
- Deck transformations.
- Simplicial homology definition and
- Singular homology definition and basic
- Exact Sequences
- Algebraic Topology, by A. Hatcher,
Chapters 1 and 2.1, 2.2 (through Mayer-Vietoris)
- Algebraic Topology, by Greenburg,
parts I and II through 17
- Key examples of manifolds such as
spheres, tori, projective spaces.
- Quotients, submanifolds, regular
level sets, Lie groups.
- Smooth maps between manifolds.
- Differential and rank of a smooth
- Regular level set theorem (Implicit
- Vector fields, integral curves.
- Lie algebra of a Lie group.
Differential Forms and Integration:
- Wedge product, pullback of forms,
- Orientation, integral of an n-form,
- An Introduction to Manifolds
(Sections 1-23) by Loring W. Tu
(Springer Universitext, 2011).
- Foundations of Differentiable
Manifolds and Lie Groups (Chapters 1-3)
by Frank Warner (Springer GTM)