PSU Mark

Mathematics Department

Graduate Program

Eberly College of Science Mathematics Department

Course Outlines

The following outlines describe first year graduate courses.

Algebra

  1. Monoids and Groups

    Isomorphism Theorems. Cayley's Theorem. Cyclic groups and their endomorphisms. Permutation groups and group actions. Homomorphisms and homomorphism theorems. Sylow's Theorems

  2. Rings

    Matrix rings and quaternions. Ideals and quotient rings. Chinese remainder theorem. Homomorphisms and homomorphism theorems. Fields of fractions. Polynomial rings. Symmetric functions. PIDs and Euclidean domains. Polynomial extensions of UFDs.

  3. Modules

    Free modules and matrices. Direct sums of modules. Finitely generated modules over a PID. Applications of Abelian groups, Diophantine equations, etc.

  4. Splitting Fields

    Galois groups. Solvable groups. Galois' criterion. General equations of nth degree. Finite fields.

The most recent text that has been used:

Qualifying exam problems from previous years

Analysis

  1. Complex Analysis

    Various forms of Cauchy's theorem. Cauchy integral formula. Power series, Laurent expansion. Residue calculus and applications. Properties of harmonic functions. Conformal mapping. Riemann mapping theorem (proof as time allows).

  2. Measure and Integration Theory

    The fundamental limit theorems. Lp spaces, elementary approximation theory. Fubini's Theorem. Elementary Fourier analysis.

  3. Functional Analysis

    The Hahn-Banach Theorem and Open Mapping Theorem. The Uniform Boundedness Principle. Hilbert Spaces. Dual Spaces. Duals of Lp spaces. Selections from the following topics as time allows: Weak and weak* topology. Banach-Alaoglu Theorem. Elementary operator theory through the spectral theorem for compact normal operators.

The most recent texts that have been used:

Tests and Homework problems from previous courses

Qualifying exam problems from previous years

Topology / Geometry

  1. General Topology

    Topological spaces and continuous mappings. Connectedness, compactness, separation. Metric spaces, criteria of metrizability. Topological groups.

  2. Algebraic Topology

    Homotopy equivalence. Fundamental group and covering spaces. Various homology theories. CW-complexes. Relation between homology and fundamental group. Locally trivial fibrations and exact sequences.

  3. Differential Topology / Geometry

    Manifolds and differentiable structures. Vector bundles. Tangent bundle. Vector fields. Riemannian metrics. Differential forms and the Poincare Lemma. Integration and Stokes Theorem. De Rham cohomology.

There is no standard text. In recent years the following texts were used for various parts of the course:

Tests and Homework problems from previous courses

Qualifying exam problems from previous years

Numerical Analysis

First Semester

  1. Polynomial interpolation and approximation ([SB] 2.1, 2.3, 2.4, [IK] 5)
    1. Orthogonal polynomials
      1. Legendre polynomials and best L_2 approximation
      2. Chebyshev polynomials
    2. Lagrange interpolation
      1. error analysis
      2. divided differences
      3. interpolation using Chebyshev points
    3. Piecewise polynomial interpolation
      1. piecewise Lagrange and Hermite interpolation
      2. spline interpolation
      3. error analysis
    4. Polynomial approximation theory
      1. Weierstrass theorem
      2. Bernstein polynomials
    5. Trigonometric interpolation and Fast Fourier Transforms
  2. Quadrature and numerical integration ([SB] 3.1-3.6, [IK] 7.0-7.5)
    1. The trapezoidal and Simpson rules, Newton-Cotes rules
    2. Euler-Maclaurin expansion
    3. Romberg integration
    4. Adaptive quadrature
    5. Gaussian quadrature
  3. Numerical linear algebra
    1. Gaussian elimination with pivoting ([SB] 4.1-4.3)
    2. Matrix transformations and special matrix forms ([SB] 6.4-6.5)
    3. Linear least squares ([SB] 4.8)
    4. Power methods ([IK] 4.2)
  4. Nonlinear systems of equations ([SB] 4.8, 5.1-5.7)
    1. Newton's and quasi-Newton's method
    2. Broyden's method
    3. Nonlinear least squares
    4. Gauss—Newton methods

Second Semester

  1. Numerical ordinary differential equations ([AI] 1-5, [SB] 7.0-7.2, [IK] 8)
    1. Euler method
    2. Multistep methods
      1. Adams-moulton, Adams-Bashforth method
      2. predictor-corrector scheme
    3. Runge-Kutta methods
    4. Stiffness
    5. Error estimation and stepsize control
  2. Numerical partial differential equations ([AI] 7, 8, 13, 14, [SB] 7.3-7.7, [IK] 9)
    1. Elliptic boundary value problems
      1. finite difference methods
      2. finite element methods
    2. Parabolic equations
      1. semi-discrete approximation
      2. convergence theory
      3. fully discrete scheme
    3. Hyperbolic equations
      1. linear hyperbolic equations
      2. stability, consistency and convergence
      3. conservation laws
  3. Sparse matrices and iterative methods ([AI] 9-11, [SB] 8)
    1. Gaussian elimination for sparse matrices
    2. Iterative methods with applications to discretizations of PDEs
      1. Jacobi, Gauss-Seidel, SOR
      2. conjugate gradient method
    3. Multigrid and domain decomposition method

Recommended texts:

Some remarks:

Qualifying exam problems from previous years

Logic and Foundations

First Semester

  1. The Propositional Calculus
    1. Boolean operations
    2. Truth assignments
    3. The tableau method
    4. The Completeness Theorem
    5. The Compactness Theorem
    6. Combinatorial Applications
  2. The Predicate Calculus
    1. Quantifiers
    2. Structures
    3. Satisfiability
    4. Tableaux
    5. The Completeness Theorem
    6. The Compactness Theorem
  3. Proof Systems for Propositional and Predicate Calculus
    1. Hilbert-style systems
    2. Gentzen-style systems
    3. The Interpolation Theorem
  4. Extensions of the Predicate Calculus
    1. Predicate calculus with identity
    2. Predicate calculus with operations
    3. Categoricity
    4. Countable categoricity
    5. Many-sorted predicate calculus
  5. Theories, Definability, Interpretability
    1. Mathematical theories (groups, fields, vector spaces, ordered structures)
    2. Foundational theories (arithmetic, geometry, set theory)
    3. Practical completeness
    4. Definability
    5. Implicit definability
    6. Beth's Theorem
    7. Interpretability
  6. Arithmetization and Incompleteness
    1. Primitive recursive functions
    2. Representability
    3. Godel numbering
    4. The Diagonal Lemma
    5. Tarski's Theorem on Undefinability of Arithmetical Truth
    6. Godel's Incompleteness Theorem
    7. Rosser's Incompleteness Theorem
    8. Godel's Theorem on Unprovability of Consistency

Second Semester

  1. Computability
    1. Primitive recursive functions
    2. The Ackerman Function
    3. Computable functions
    4. Partial recursive functions
    5. The enumeration theorem
    6. The halting problem
    7. Examples of functions and sets which are not computable
  2. Undecidability of the Natural Number System
    1. Terms
    2. Formulas
    3. Sentences
    4. Arithmetical definability
    5. Chinese remainder theorem
    6. Definability of computable functions
    7. Definability of the halting problem
    8. Godel numbers
    9. Undefinability of arithmetical truth
  3. Decidability of the Real Number System
    1. Effective functions
    2. Quantifier elimination (P. J. Cohen's method
    3. Definability over the real number system
    4. Decidability of the real number system
    5. Decidability of Euclidean geometry
  4. Introduction to Set Theory
    1. Russell paradox
    2. Operations on sets
    3. Cardinal numbers
    4. Ordinal numbers
    5. Transfinite recursion
    6. The Axiom of Choice
    7. The Well Ordering Theorem
    8. The Continuum Hypothesis
    9. Measurable cardinals
  5. Independence of the Continuum Hypothesis
    1. The Zermelo-Fraenkel axioms
    2. Set-theoretic foundations of mathematics
    3. Models of set theory
    4. Inner models
    5. Constructible sets
    6. The inner model L
    7. The generalized continuum hypothesis in L
    8. Models constructed by forcing
    9. A model where the continuum hypothesis fails

Recommended texts:

Some remarks:

Tests and Homework problems from previous courses

Qualifying exam problems from previous years

Partial Differential Equations

  1. Classical linear equations: transport, Laplace, heat, and wave equations
    1. basic properties
    2. fundamental solutions
    3. mean value properties
    4. maximum principles
    5. energy methods
    6. Fourier transform method
  2. First order nonlinear PDE's
    1. characteristics
    2. conservation laws
    3. shocks
  3. Special solutions
    1. similarity solutions
    2. traveling waves
    3. power series methods
  4. Sobolev spaces
    1. distributions
    2. weak derivatives
    3. weak convergence
  5. More on Sobolev spaces
    1. traces
    2. Poincaré
    3. Sobolev inequalities
    4. embeddings
  6. Second order elliptic equations
    1. fixed point theorems
    2. weak solutions
    3. regularity
    4. maximum principles
    5. eigenvalues
    6. applications
  7. Evolution equations
    1. weak solutions
    2. Galerkin method
    3. regularity
    4. maximum principles and propagation of disturbance
    5. applications
  8. Semigroup theory
    1. infinitesimal generators
    2. Hille-Yoshida theorem
    3. applications
  9. Calculus of variations and its applications
    1. direct method
    2. Mountain pass theorem

The above consists of the core part of the first-year graduate study on the subject of Partial Differential Equations at PSU. Each instructor may add a few additional topics. Math 513-4 is a year long course covering the above and provide an introduction to the fundamental theories and methods in partial differential equations. The first course M513 will cover topics 1-4 listed above and the second course M514 will cover the rest. Most of the first 5 chapters of Evans' book will be covered in M513.

Additional references:

To pursue a Ph.D. in the area of PDE, we recommend that you keep in mind ODE is an important component. PSU has M411 (ODE/Fourier Series), M412 (PDE), and M511 (ODE) before the course level of M513-4. After M513-4, PSU offers two topics courses per semester in the broad area of PDE, Numerical Analysis, and Applied Mathematics. While you are studying M513-4 in the first year, you can attend lower level courses or upper level topics courses depending on your preparedness.

Qualifying exam problems from previous years