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Graduate Program

Qualifying Exams

Doctoral students must take quals in three of the six available areas: (1) Analysis, (2) Geometry, (3) Topology, (4) Algebra, (5) PDEs, and (6) Numerical Analysis. Please see the Graduate Handbook for more details on the exams, when they will be offered, and how to sign up to take them.

Core Examination Topics

(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, \(1\le p<\infty\))

References

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

(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 n-form, Stokes' theorem.

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 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
  • Mayer-Vietoris

References

  • 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

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 (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.

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

(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

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

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)