Tufts University
Department of Mathematics

The Qualifying Examinations

Analysis Core Examination Topics

  • Real Analysis

  • Metric and topological spaces: Completions, contraction mapping principle, product spaces, connectedness, compactness, bases and countability.
  • Measure and integration: Lebesgue measure in Rn, convergence theorems, Radon--Nikodym theorem, Fubini's theorem.
  • Linear spaces: Banach and Hilbert spaces, linear operators, Lp-spaces, Riesz representation theorem.

  • Complex Analysis

  • Complex functions: Analyticity, Cauchy--Riemann equations, singularities, Laurent series, uniform convergence.
  • Complex integration: Cauchy--Goursat theorem, Cauchy integral formula, Morera's theorem, Liouville's theorem, maximum modulus principle, residue theorem, definite integrals.

    • References

  • Real Analysis by H. L. Royden (Macmillan)
  • Real and Complex Analysis by Walter Rudin (McGraw-Hill)
  • Complex Variables and Applications by Ruel V. Churchill (McGraw-Hill)

    • Geometry Core Examination Topics

  • 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 n-form, Stokes' theorem.

  • Fundamental Group and Covering Spaces:

  • Van Kampen's theorem.
  • Classification of covering spaces.
  • Deck transformations and group actions.

    • References

  • An Introduction to Manifolds (Chapters 1-22) by Loring W. Tu (Springer Universitext)
  • Foundations of Differentiable Manifolds and Lie Groups (Chapters 1-3) by Frank Warner (Springer Graduate Texts in Mathematics)
  • A Basic Course in Algebraic Topology (Chapters 2-5) by William S. Massey (Springer Graduate Texts in Mathematics)

    • Algebra Core Examination Topics

  • 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≥ 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 (Addison-Wesley)
  • Algebra by Michael Artin (Prentice Hall)
  • Abstract Algebra by David S. Dummit and Richard M. Foote (Prentice Hall)

    • Applied Core Examination Topics

    Note: An individual applied examination will cover either PDE or numerical analysis.

    Partial Differential Equations

  • Linear Partial Differential Equations:

  • Elliptic PDE: Laplace, Poisson and Helmholtz equations, boundary-value problems, existence and uniqueness, weak and strong elliptic maximum principles, boundary regularity.
  • Parabolic PDE: Heat equation, Schrödinger equation, existence and uniqueness of solutions, weak and strong parabolic maximum principles, regularity, weak solutions, Lax-Milgram Theorem, Galerkin method.
  • 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:

  • Hyperbolic systems, shallow-water equations, gas-dynamic equations, Fourier methods, energy methods.
  • Method of characteristics, weak solutions, jump conditions, entropy conditions.

    • References

  • Partial Differential Equations: Methods and Applications by R. C. McOwen (Pearson Prentice-Hall)
  • Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative PDEs and the Theory of Global Attractors (Parts I and II) by J. C. Robinson (Cambridge University Press)
  • Basic Linear Partial Differential Equations by F. Trèves (Academic Press, Dover reprint)
  • An Introduction to Nonlinear Partial Differential Equations by J. D. Logan (Wiley)

    • Numerical Analysis

  • 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.
  • Nonlinear Systems of Equations, Optimization: Newton's method, quasi-newton methods, fixed point iteration. Newton and Levenberg-Marquardt methods for unconstrained optimization.

  • Numerical Approximation

  • Interpolation: Lagrange and Hermite interpolating polynomials, Runge phenomena. Splines, least squares approximation of functions and orthogonal polynomials.
  • Integration: 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, Method of lines, Finite difference schemes for the one-way wave equation (upsinding, Lax-Friedrichs, Lax-Wendroff), Flux limiters.

    • 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 (Click Here to Download)
  • 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)