Tufts University  |  School of Arts and Sciences  |  School of Engineering  |  Find People  | 

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

Metric and topological spaces:

Topologies, bases, ways of generating topologies, Hausdorff, separable, first and second countable, Completion of metric spaces, complete metric spaces (Baire category theorem), Compactness in arbitrary topological spaces, in metric spaces, in \((C(K); d_{\sup} )\) with \(K\) compact (Arzela-Ascoli theorem), in product spaces with the product topology (Tychonoff's theorem)

Measure and integration:

General measure spaces (including sigma-algebras, measures), Caratheodory Extension theorem, Borel sets, Lebesgue measurable sets, counting measure, Cantor set and Cantor function, Borel measures on the real line, measurable functions, non-Lebesgue measurable set, completion of a measure, Lebesgue integral, relation of Riemann integral to the Lebesgue integral, Limit theorems for integrals (monotone and dominated convergence theorems, Fatou's Lemma), Absolutely continuous and singular measure (Radon-Nikodym theorem), Product measures on product-measurable spaces, Fubini-Tonelli theorem.

Banach spaces:

\(L_p(X; S; \cdot), 1 \le p < \infty\) and \(l^p\)-Holder's and Minkowski's inequalities, relationships between \(L_p\) spaces for different \(p\), Hilbert spaces (inner products, orthogonal decomposition, orthonormal bases) \((C(K); \|\cdot\|_\infty)\), Bounded linear operators (dual spaces including Riesz representation theorems for Hilbert spaces, \(L_p\)-spaces, \(1\le p<\infty\))


  • Real Analysis by H. L. Royden (Macmillan)
  • Real and Complex Analysis by Walter Rudin (McGraw-Hill)

(2) Geometry


  • 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 n-form, Stokes' theorem.


  • 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)

(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.


  • Simplicial homology definition and computations.
  • Singular homology definition and basic properties.
  • Exact Sequences
  • Degree
  • Mayer-Vietoris


  • 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

(4) Algebra


  • Quotients and Isomorphism Theorems for groups, rings, and modules.


  • 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).


  • 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.


  • 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 (Prentice Hall)

(5) Partial Differential Equations

Linear Partial Differential Equations

Elliptic PDE:
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.

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, 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 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, Gauss-Seidel, 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, quasi-newton methods, fixed point iteration. Newton and Levenberg-Marquardt methods for unconstrained optimization.

Numerical Approximation

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.

Differential Equations:
Convergence of explicit one-step methods, Stiffness, A- stability, impossibility of A-stable explicit Runge-Kutta methods


  • 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)