Fractal Geometry

Speaker

Professor Sze-Man Ngai

Time

Fridays, 9:50 am-12:15

Feb. 21 - Jun. 6, 2025

Venue

C654

Shuangqing Complex Building A

Course description

Fractals are highly non-smooth 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. The first part of this course (10 weeks) covers the following core topics: Hausdorff measure and dimension, packing measure and dimension, box dimension, the collage theorem, iterated function systems, self-similar sets, the Moran-Hutchinson theorem, self-similar measures. The second part of this course (6 weeks) will survey several research-oriented topics. Based on students' interests, the topics will be selected from the following list: the multifractal formalism, Hausdorff dimension of self-similar sets with overlaps, the Laplacian and analysis on fractals, self-affine tilings, complex dynamical systems and the Mandelbrot set, and fractals in Riemannian manifolds.

Prerequisite:

undergraduate analysis

Speaker

Sze-Man Ngai

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, and Georgia Institute of Technology. He joined the Department of Mathematical Sciences of Georgia Southern University in 2000.

His main research areas are fractal geometry and the theory of fractal measures. Because of the underlying self-similarity, he is also interested in the theories of wavelets, self-affine tiles, fractal differential equations, and Markov chains and boundary theory on fractals.