Graduate Program
Qualifying Exams
Doctoral students must take quals in three of the six available areas:
(1) Analysis, (2) Geometry, (3) Algebraic Topology, (4) Algebra, (5)
PDEs, and (6) Numerical Analysis. 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.
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) Analysis
In addition to the topics covered below, students are
expected to be proficient in the topics covered in a
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)
(2) 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)
(3) 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
(4) 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 \(n\ge 5\).
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)
(5) 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)
(6) Numerical Analysis and Numerical 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)
