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 ${\Bbb R}^n$, convergence theorems, Radon--Nikodym theorem, Fubini's theorem.
  • Linear spaces: Banach and Hilbert spaces, linear operators, $L^p$-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

  • Advanced multivariate calculus:
  • Differential calculus: Derivatives as linear maps, inverse and implicit function theorems.
  • Integral calculus: Differential forms, Stokes' theorem, Poincar\'e lemma.
  • Structure of manifolds and varieties: Coordinate systems, tangent vectors, tangent spaces, differentials, vector fields, classification of compact (closed) surfaces. Students should be comfortable with key examples of manifolds such as $n$-spheres (up to $n=3$), the 2-torus, and some elementary Lie groups and their Lie algebras.
  • Basic topological constructs:
  • Deformations of curves and maps, fundamental group.
  • Chain complexes and homology theory, primarily for surfaces and simple higher-dimensional manifolds.
  • Covering spaces, universal covers of surfaces.
  • Basic geometric constructs: Riemannian metrics, geodesics, geodesic completeness.
  • References:
  • Lecture notes on elementary topology and geometry by Isadore M. Singer and John A. Thorpe (Scott, Foresman)
  • Foundations of Differentiable Manifolds and Lie Groups (Chapters 1, 2) by Frank Warner (Springer-Verlag)
  • Analysis on Manifolds by James R. Munkres (Addison-Wesley)
  • Calculus on Manifolds by Michael Spivak (W. A. Benjamin)
  • Riemannian Geometry by Manfredo do Carmo (Birkh\"auser)
  • Elements of Differential Geometry by Richard Milman and George Parker (Prentice-Hall)
  • Basic Topology by M. A. Armstrong (Springer-Verlag)
  • Elements of Algebraic Topology by James R. Munkres (Prentice Hall)

    • Algebra core exam topics

  • Groups: Permutation groups, homomorphism and isomorphism theorems, Sylow theorems, solvable groups and nonsolvability of symmetric groups.
  • Rings and Integral Domains: Fields of fractions, Euclidean domains, principal ideal domains, unique factorization domains and polynomials over them.
  • Modules: Free, torsionfree, and torsion modules, finitely generated modules over a principal ideal domain, applications to classification of abelian groups and calculation of canonical forms of linear transformations, tensor products.
  • Fields: Algebraic, transcendental, separable, and normal extensions, splitting fields, Galois theory of finite extensions in arbitrary characteristic, geometric constructibility, finite fields, algebraic closures.
  • 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)
    • Return to Information for Prospective Graduate Students.
    Back