Course Outlines

The following outlines describe first year graduate courses.

• Algebra
• Analysis
• Topology / Geometry
• Numerical Analysis
• Logic and Foundations
• Partial Differential Equations

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:

• Basic Algebra I, 2nd Edition by N. Jacobson

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:

• Real & Complex Analysis, 3rd Edition by W. Rudin
• Functional Analysis, 3rd Edition by W. Rudin
• Measure Theory by P. Halmos

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:

• Introduction to Algebraic Topology by Rotman
• Differential Geometry by S. Sternberg

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:

• [AI]: A First Course in the Numerical Analysis of Differential Equations, by Irieh Iserles, Cambridge university Press 1996.
• [IK]: Analysis of Numerical Methods, by E. Isaacson and H. B. Keller, Wiley 1966 (or Dover 1994).
• [SB]: Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch, 2nd edition, Springer Verlag 1993.

Some remarks:

• This is the syllabus for two numerical analysis graduate course sequence, 6 credits.
• The three references will be put on reserve in the math library. Relevant sections from these books are noted in the parentheses.
• The course will initially be offered every other year, starting from Fall 1998.
• The qualifying exam for this topic will cover a broad range of topics, including theory, algorithm and implementation. The registration numbers of these two courses are yet to be assigned.

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:

• Raymond Smullyan, First-order Logic, Springer-Verlag.
• Herbert Enderton, A Mathematical Introduction to Logic, Academic Press.
• Elliott Mendelson, Introduction of Mathematical Logic, 3rd edition, Wadsworth.
• Joseph R. Shoenfield, Mathematical Logic, Addison-Wesley.
• Moshe Machover and John Bell, A Course in Mathematical Logic, North-Holland.
• Hartley Rogers, Theory of Recursive Functions and Effective Computability, MIT Press.
• Kenneth Kunen, Set Theory, North-Holland.
• Thomas Jech, Set Theory, Academic Press.

Some remarks:

• This is the syllabus for the basic two-semester graduate course sequence in Logic and Foundations, Math 557-558, 6 credits.

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.

• Text 1: Partial Differential Equations by L. C. Evans
• Text 2: Introduction to Partial Differential Equations, G. B. Folland, 2nd Ed (1995)

Additional references:

• Partial Differential Equations, R. McOwen
• Hilbert Space Methods for Partial Differential Equations, R. Showalter (available free electronically)
• Elliptic Partial Differential Equations of Second Order, D. Gilbarg and N. S. Trudinger, 2nd Ed or later.
• Partial Differential Equations, F. John

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