This course covers the theoretical underpinnings of real analysis. It is intended for students planning to go on to graduate school in mathematics. Stress will be placed on students proving theorems correctly. Homework will be collected weekly and graded. As time permits, we will provide examples giving application of the theory to problems in analysis.

Topics:
Math 4331
1. The Real and Complex Number Systems
2. Basic Topology, Metric Spaces
3. Numerical Sequences and Series
4. Continuous Functions
5. Derivatives
Note that topics 3-5 will be covered in considerably more depth than one
encounters in a Calculus course.

Math 4332 (next semester)
1. The Riemann-Stieltjes Integral
2. Sequences and Series of Functions
3a. Exponential and Trigonometric functions
3b. Fourier Series
3c. The Gamma Function
4. Functions of Several Variables:
4a. Linear Transformations
4b. Derivatives
4c. The Contraction Mapping Theorem
4d. The Inverse Function and Implicit Function theorems
4e. Differentiation of Integrals
5. The Lebesgue Theory of Integration

The location of the boundary between 4331 and 4332 is somewhat uncertain.

The material for the course is a subset of the material listed in the text above. More specifically, we will cover the sections below.

Chapter 1: Where PDEs come from

1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems

Chapter 2: Waves and Diffusions

2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions

Chapter 4: Boundary Problems

4.1 Separation of Variables, the Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition

Chapter 5: Fourier Series

5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and General Fourier Series
5.4 Completeness

Chapter 6: Harmonic Functions

6.1 Laplace's Equation
6.2 Rectangles and Cubes
6.3 Poisson's Formula

Regular homework will be given, along with a midterm exam and a final exam.
There will be an online live lecture each week on Thursdays from 4-6pm. This
lecture will be recorded and available on the course homepage.

Grades: 30% midterm exam, 40% final exam, 30% homework.

A link to the course homepage will appear at http://www.math.uh.edu/~jmorgan prior to the beginning of the fall 2012 semester.

Partial Differential Equations, An Introduction, by Walter A. Strauss.

This course covers the theoretical underpinnings of real analysis. It is intended for students planning to go on to graduate school in mathematics. Stress will be placed on students proving theorems correctly. Homework will be collected weekly and graded. As time permits, we will provide examples giving application of the theory to problems in analysis.

Topics:
Math 4331
1. The Real and Complex Number Systems
2. Basic Topology, Metric Spaces
3. Numerical Sequences and Series
4. Continuous Functions
5. Derivatives
Note that topics 3-5 will be covered in considerably more depth than one
encounters in a Calculus course.

Math 4332 (next semester)
1. The Riemann-Stieltjes Integral
2. Sequences and Series of Functions
3a. Exponential and Trigonometric functions
3b. Fourier Series
3c. The Gamma Function
4. Functions of Several Variables:
4a. Linear Transformations
4b. Derivatives
4c. The Contraction Mapping Theorem
4d. The Inverse Function and Implicit Function theorems
4e. Differentiation of Integrals
5. The Lebesgue Theory of Integration

The location of the boundary between 4331 and 4332 is somewhat uncertain.

This course meeting is online or in classroom.

This course treats topics related to the solvability of various types of equations, and also of optimization and variational problems. The first half of the semester will concentrate on introductory material about norms, Banach and Hilbert spaces, etc. This will be used to obtain conditions for the solvability of linear equations, including the Fredholm alternative. The main focus will be on the theory for equations that typically arise in applications. In the second half of the course the contraction mapping theorem and its applications will be discussed. Also, topics to be covered include finite dimensional implicit and inverse function theorems, and existence of solutions of initial value problems for ordinary differential equations and integral equations.

This course meeting is online or in classroom.

The most natural question for a student before joining this class usually is: What is an inverse problem? It is the aim of this class to introduce the students to the mathematical and physical principles behind various inverse problems and to discuss how they can be used to understand questions from medical imaging, radar imaging, cloaking and invisibility, oil explorations and recovery, shape optimization or recovering the inner structure of the earth.

A very non-rigorous definition of an inverse problem is that, in a sense, an inverse problem is the “dual” to a direct problem associated to the same model. The simplest example of such dual pair, i.e., direct/inverse problems, is associated to a linear system: here the direct problem will be, for a given mxn matrix A and a given nx1 vector v determine the product Av while the inverse problem will be, for known Av, determine v. So every time you solved a linear system you solved in fact an inverse problem.

For medical imaging, the practical question will be to detect cancerous tumors, or other abnormalities present in a part of the human body by non-invasive methods, i.e., by using only data collected without direct contact with the patient. Thus, from the acquired data the scientist would like to invert and obtain information about the source of this data. The same scenario is valid for the problem of mapping the inner structure of the earth or the problem of oil exploration and recovery problems, where the engineer would like to be able to map the inner structure of the earth (thus detecting potential oil resources and their size) only from inexpensive data acquired at the surface of the earth. For the shape optimization problem the practical question may be, for example, to find the shape that globally minimizes cost and weight, still giving optimal performance.

The cloaking question is, as the name suggests, related to the desire of making certain targets invisible to radar or sonar. A possible inverse problem approach to this question will inquire into the possibility to design suitable antennas to achieve the desired nulling around the object of interest while maintaining its communication capability.