Introduction to Fractal Geometry
Fractals are highly nonsmooth sets that often possess self-similarity, infinite irregularity, and non-integral Hausdorff dimension. They arise naturally in many branches of mathematics, science, and engineering. This is an introductory course on fractals. Topics include Hausdorff measure and dimension, box dimension, iterated function systems, self-similar sets, the collage theorem, the Moran-Hutchinson theorem, self-similar measures, and the multifractal formalism. Finally, a survey on iterated function systems with overlaps will be given; tentatively, topics will be selected from the weak separation property, the finite type condition, infinite Bernoulli convolutions and related results by Solomyak, Hochman, Varju, and others.
讲师
日期
2024年09月10日 至 12月17日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 09:50 - 11:25 | A7-101 | ZOOM 11 | 435 529 7909 | BIMSA |
修课要求
Undergraduate analysis
课程大纲
Chapter 1. Introduction and preliminaries
Chapter 2. Hausdorff measure and Hausdorff dimension
Chapter 3. Box dimension
Chapter 4. Some techniques for calculating dimension
Chapter 5. Iterated function systems and self-similar sets
Chapter 6. Applications to encoding images and the collage theorem
Chapter 7. Self-similar measures
Chapter 8. Survey of iterated function systems with overlaps
Homework assignments will be given regularly but will not be collected for grading.
Chapter 2. Hausdorff measure and Hausdorff dimension
Chapter 3. Box dimension
Chapter 4. Some techniques for calculating dimension
Chapter 5. Iterated function systems and self-similar sets
Chapter 6. Applications to encoding images and the collage theorem
Chapter 7. Self-similar measures
Chapter 8. Survey of iterated function systems with overlaps
Homework assignments will be given regularly but will not be collected for grading.
参考资料
Kenneth Falconer, Fractal Geometry, Mathematical Foundation and Applications, 3rd ed, Wiley, 2014. ISBN: 978-1-119-94239-9.
听众
Undergraduate
, Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
不公开
语言
英文
讲师介绍
Dr. Ngai received his B.Sc. from University of Hong Kong, and his M.A. and Ph.D. from University of Pittsburgh. After receiving his Ph.D., he has held research and teaching positions at The Chinese University of Hong Kong, Cornell University, Georgia Institute of Technology, and Georgia Southern University. He joined Beijing Institute of Mathematical Sciences and Applications as a professor in 2024. His main research areas are fractal geometry and the theory of fractal measures. He is also interested in the theories of wavelets, self-affine tiles, fractal differential equations, and spectral graph theory.