This book is written for mathematics students who have encountered basic complex analysis and want to explore more advanced project and/or research topics. It could be used as (a) a supplement for a standard undergraduate complex analysis course, allowing students in groups or as individuals to explore advanced topics, (b) a project resource for a senior capstone course for mathematics majors, (c) a guide for an advanced student or a small group of students to independently choose and explore an undergraduate research topic, or (d) a portal for the mathematically curious, a hands-on introduction to the beauties of complex analysis. Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on student-driven and technology-enhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and hands-on manipulation. There are more than 15 Java applets that allow students to explore the research topics without the need for purchasing additional software.
Table of Contents
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
About the Authors
About the Authors
Michael Brilleslyper was raised in southern California. He earned a bachelor’s degree in applied mathematics and a master’s degree in mathematics from Arizona State University. He obtained his Ph.D. from the University of Arizona in 1994, under the direction of Doug Pickrell. He spent the first six years of his career at Arizona State working extensively with their introductory and calculus courses. For two of those years he served as coordinator for their First Year Mathematics Program. In 2000, he came to the United States Air Force Academy in Colorado Springs, CO. At the Academy, Dr. Brilleslyper enjoys teaching at all levels of the curriculum. He has worked on numerous curricular development projects involving writing, technology, and fundamental skills. He is extremely active in the MAA, having been in the second cohort of Project NExT fellows. More recently he has served as Rocky Mountain section chairman, he has twice been a program chair, he has served as chair of the professional development committee, and he currently serves as Governor of the Rocky Mountain section. He lives in Colorado Springs with his wife and their two daughters.
Michael Dorff is a professor of mathematics at Brigham Young University. He was born in Duluth, Minnesota and grew up in southern California. After teaching high school for four years, he earned an MS degree at the University of New Hampshire and in 1997 a Ph.D. from the University of Kentucky in complex analysis. He was a professor at the University of Missouri - Rolla before accepting a position in 2000 at BYU. He has published over 20 research papers and has given talks at over 100 different conferences, universities, and colleges. He founded the BYU mathematics REU and in 2007 he founded the national Center of Undergraduate Research in Mathematics, which promotes, trains, and supports professors across the U.S. in doing research with undergraduate students. He is a member of the MAA, AMS, CUR, and Project NExT, and has served in many positions including governor of the MAA Intermountain section, chair of several MAA committees, member of the Executive Board of CUR, and member of the editorial boards of the journals American Mathematical Monthly and Involve. He was a Fulbright Scholar in Poland and has received numerous teaching awards including the MAA’s Deborah and Franklin Tepper Haimo Award in 2010. He is married with 5 daughters. His interests include reading (Dostoyevsky and Dickens through Stegner and Saramago), traveling (invite him to visit you!), running (even at 3 am on the streets in Utah), music (classical, Norah Jones), and soccer.
Jane McDougall received her Ph.D. from Northwestern University in 1996, where she studied functions of one complex variable. She has been a member of the Department of Mathematics and Computer Science at Colorado College since 1997, during which time her research interests have grown into geometric function theory and harmonic mappings.
Jim Rolf was born in Nashville, Tennessee but moved to his adopted home state of Texas at age two. He earned his Bachelor of Science degree in mathematics in his home town of Waco, Texas while attending Baylor University. This was followed by a Master of Divinity degree from Southwestern Baptist Theological Seminary in Fort Worth, Texas. In 1997 he completed his doctoral work in numerical analysis at Duke University under the direction of William K. Allard. After three years on the faculty at West Point, Jim moved to Colorado Springs, CO and the math department at the United States Air Force Academy. In the fall of 2012, Jim will join the math department at Yale University. In addition to his professional interests, Jim enjoys cycling, gardening, and woodworking.
Lisbeth Drews Schaubroeck was born in the Netherlands, but spent most of her childhood in Geneseo, Illinois. She earned her bachelor’s degree in Mathematics and Mathematics Education at Wartburg College in Waverly, Iowa. In 1998, she completed her doctoral work in complex analysis at the University of North Carolina at Chapel Hill under the direction of John Pfaltzgraff. Currently, Beth is a professor at the United States Air Force Academy in Colorado Springs, Colorada. There she enjoys teaching all levels of undergraduate mathematics to future Air Force officers. She is active in the Rocky Mountain Section of the MAA and has been on the awards committee and the student activities coordinator. Beth especially enjoys her work in faculty development in a wide range of venues. She mentors Air Force officers who are new to teaching at the Air Force Academy, new Ph.Ds through Project NExT, and seasoned faculty from civilian universities and community colleges through an NSF-funded PREP workshop. She lives in Colorado Springs with her husband and two sons.
Richard L. Stankewitz was born in Royal Oak, Michigan. He earned a bachelor’s degree with distinction in mathematics from the University of Michigan at Ann Arbor in 1991. He obtained a Ph.D. in 1998 under the direction of his advisor Aimo Hinkkanen at the University of Illinois at Urbana in the field of complex dynamics. Dr. Stankewitz held positions at Texas A&M University at College Station and Penn State University at Erie before coming to Ball State University in 2002, where in 2010 he attained his current position of Professor of Mathematics. He is a member of the AMS and has a long time interest in supporting efforts to encourage undergraduate research in mathematics. In addition to publishing over 15 research articles, he has given talks in over 50 national and international conferences, universities, and colleges.
Ken Stephenson was born in South Haven, Michigan, in 1945. He received a BS in mathematics at Michigan and an MS in mathematics at Wisconsin before serving 3 years as a Naval Officer. He returned to his studies at Wisconsin, receiving his Ph.D. under Walter Rudin in 1976. From Wisconsin he joined the faculty at the University of Tennessee, where he continues his research and teaching. Complex function theory and the associated conformal geometry have remained Dr. Stephenson’s core mathematical interest, but in 1985 a fascinating talk by Bill Thurston on “circle packing” profoundly changed his outlook. He began contributing to the development of circle packing and its associated “discrete” conformal geometry through publications and software, culminating in his 2005 book, Introduction to Circle Packing: the Theory of Discrete Analytic Functions, Cambridge University Press, and his software package “CirclePack”. This is a new type of experimentally driven mathematics, and his principal goal now is promoting circle packing and its applications. Dr. Stephenson has been supported throughout his years of research by grants from the National Science Foundation, the Tennessee Science Alliance, and currently the Simons Foundation. He has spoken on circle packing at numerous national and international conferences and has held visiting positions at the University of Hawaii, the Open University (England), the University of Cambridge, Florida State University, the Technical University of Berlin, the Free University of Berlin, Oak Ridge National Laboratory, and the Mathematical Sciences Center (Tsinghua University, Beijing). He is a member of the American Mathematical Society and the Mathematical Association of America, and is a Fellow of AAAS (American Association for the Advancement of Science).