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Graduate Program

Qualifying Exams

Doctoral students must take quals in three of the five available areas: (1) Analysis, (2) Geometry/Topology, (3) Algebra, (4) PDEs, and (5) Numerical Analysis. Within the Geometry/Topology area, students must choose Geometry or Topology.

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)

(2a) 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)

(2b) 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

(3) 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)

(4) 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, weak solutions, Lax-Milgram Theorem, Galerkin method.

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:

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. Trèves (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

(5) 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
  • 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)