This new offering from the MAA is a collection of six inducements or invitations to further research for undergraduates with some background in complex analysis. The book is flexible enough to be a source of enrichment material, a basis for research projects, the kernel of a capstone course, or just a tool to ignite the interest of the mathematically curious.
Although the authors of the six chapters vary, the style and approach is much the same throughout. The goal is a kind of guided research that is focused on fostering independent student investigation and discovery. Each chapter begins with a guided tour of the topic and then offers students several opportunities to investigate further. Interspersed throughout the text are examples, Java applets, exercises, explorations and a variety of potential projects. The exercises are integrated into the text, designed with clear goals, and identified as essential for comprehending the material. The “explorations” are less goal-directed and aimed more at getting students to find directions to investigate on their own. The projects are optional activities, large and small, that might last a few weeks or a whole term.
The book’s six chapters offer what the authors call “current research topics”, although some are more current than others. Topics come in a reasonable variety. For those favoring geometry, there are chapters on minimal surfaces (“Soap Films, Differential Geometry and Minimal Surfaces”) and “Circle Packing” (configurations of circles with specified patterns of tangency). To those who are inclined to complex function theory, there are “Anamorphosis, Mapping Problems and Harmonic Univalent Functions” (perhaps the closest to a bona fide current research topic) and “Mappings to Polygonal Domains” (creating univalent functions from one such domain to another). For the application-minded, there is “Applications to Flow Problems”, about two-dimension vector fields in, for example, electromagnetic or fluid dynamics. Finally, there is “Complex Dynamics”, which introduces chaos and fractals via iteration of complex analytic functions.
This is an attractive book that should have a lot of appeal to students. It offers a number of excellent avenues into research for undergraduates.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
Using Java Applets
Using Links in the Electronic Book
1. Complex Dynamics: Chaos, Fractals, the Mandelbrot Set, and More 1
2. Soap Films, Differential Geometry, and Minimal Surfaces
3. Applications to Flow Problems
4. Anamorphosis, Mapping Problems, and Harmonic Univalent Functions
5. Mappings to Polygonal Domains
6. Circle Packing
The Riemann Sphere