Graduate Program
Qualifying Exams
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
PhD Handbook
for more details on the 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 2week mark and does
not show for the exam, will receive a No Pass.
Register for the next round of qualifying exams >
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, GaussSeidel, Successive Overrelexation
method), Conjugate Gradient Method
Eigenvalue Problems
Gerschgorin theorem, power method, inverse power method,
stability of eigenvalue problems
Nonlinear Systems of Equations, Optimization:
Newton's method, quasinewton methods, fixed point
iteration. Newton and LevenbergMarquardt methods for
unconstrained optimization.
Numerical Approximation
Interpolation:
Lagrange and Hermite interpolating polynomials, Runge
phenomena. Splines, least squares approximation of
functions and orthogonal polynomials.
Integration:
NewtonCotes methods, Gaussian quadrature, EulerMacLaurin
formula, Adaptive quadrature.
Differential Equations:
Convergence of explicit onestep methods, Stiffness, A
stability, impossibility of Astable explicit
RungeKutta methods
References
 An Introduction to Numerical Analysis by K. E.
Atkinson (Wiley)
 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
University Press)
(2) Analysis
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
Tufts.
Metric Spaces: StoneWeierstrass theorem,
ArzelaAscoli theorem, Baire category theorem.
Measure and Integration: General measure spaces
(including sigmaalgebras, 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 RadonNikodym theorem. Product measures
and FubiniTonelli theorem.
Functional Analysis: Banach spaces, Lp spaces,
the Holder and Minkowski inequalities, the BanachSteinhaus,
Open Mapping, and HahnBanach 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, CauchyRiemann
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.
Suggested References:
 Real Analysis by H. L. Royden
(Macmillan)
 Real and Complex Analysis by Walter
Rudin (McGrawHill)
 Real Analysis: Measure Theory,
Integration, and Hilbert Spaces by E. M.
Stein and R. Shakarchi (Princeton University
Press)
 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
University Press)
(3) Algebra
Generalities:
 Quotients and Isomorphism Theorems for groups, rings, and
modules.
Groups:
 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).
Fields:
 Algebraic, transcendental, separable, and Galois extensions,
splitting fields.
 Finite fields, algebraic closures.
 The fundamental theorem of Galois theory for a finite
extension of a field of arbitrary characteristic.
References
 Basic Algebra I by Nathan Jacobson (W. H. Freeman)
 Algebra by Thomas W. Hungerford (Springer)
 Algebra (in part) by Serge Lang (AddisonWesley)
 Algebra by Michael Artin (Prentice Hall)
 Abstract Algebra by David S. Dummit and Richard M. Foote
(Prentice Hall)
(4) Partial Differential Equations
Linear Partial Differential Equations
Elliptic PDE:
Laplace, Poisson and Helmholtz equations, boundaryvalue
problems, existence and uniqueness, Fredholm alternative, weak
and strong elliptic maximum principles, boundary regularity,
Sobolev spaces, weak solutions, LaxMilgram Theorem, Galerkin
method, variational principles, Fréchet derivative.
Parabolic PDE:
Heat equation, Schrödinger equation, existence and uniqueness of
solutions, weak and strong parabolic maximum principles,
regularity.
Hyperbolic PDE:
Wave equation, method of characteristics.
Spectral Analysis:
Fourier series, Fourier transforms, convergence and
approximation properties, generalized functions, distributions.
Eigenfunction expansion, SturmLiouville Theory, Rayleigh
quotient, RayleighRitz method, Green's functions.
Quasilinear and Nonlinear PDE:
CauchyKovalevskaya theorem, hyperbolic systems, shallowwater
equations, gasdynamic equations, Fourier methods, energy
methods. Method of characteristics, weak solutions, jump
conditions, entropy conditions.
References
 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
and Rubenstein
 Applied Function Analysis by Griffel
 Partial Differential Equations by L. C. Evans (AMS, Second
Edition)
(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.
Covering Spaces:
 Basic lifting properties.
 Universal covering.
 Relation between coverings and subgroups of
the fundamental group.
 Group actions.
 Deck transformations.
Homology:
 Simplicial homology definition and
computations.
 Singular homology definition and basic
properties.
 Exact Sequences
 Degree
 MayerVietoris
References
 Algebraic Topology, by A. Hatcher,
Chapters 1 and 2.1, 2.2 (through MayerVietoris)
 Algebraic Topology, by Greenburg,
parts I and II through 17
(6) Geometry
Manifolds:
 Key examples of manifolds such as
spheres, tori, projective spaces.
 Quotients, submanifolds, regular
level sets, Lie groups.
 Smooth maps between manifolds.
Tangent Spaces:
 Differential and rank of a smooth
map.
 Regular level set theorem (Implicit
function theorem).
 Vector fields, integral curves.
 Lie algebra of a Lie group.
Differential Forms and Integration:
 Wedge product, pullback of forms,
exterior derivative.
 Orientation, integral of an nform,
Stokes' theorem.
References
 An Introduction to Manifolds
(Sections 123) by Loring W. Tu
(Springer Universitext, 2011).
 Foundations of Differentiable
Manifolds and Lie Groups (Chapters 13)
by Frank Warner (Springer GTM)
