Title: Fractal Sets and Harmonic Analysis
Description: In the classical setting, we understand the notion of dimension as an integer determined by the "number of independent directions" one could "walk" along. However, lately, there have been several other notions of dimension introduced into Mathematics with the purpose of understanding sets that are in a sense "sparse" (or exceptional) but have a high structure. The simplest example of a fractal set is the 1/3 Cantor set, which is seemingly very small (it has one-dimensional volume zero) but is uncountable. Further, the 1/3 Cantor set has also a "self-similar" structure in the sense that if one "zooms in" on a piece of the Cantor set, it still looks like the full Cantor set.
Over the past decades, there have been a lot of advances in the intersection of (Harmonic) Analysis and Fractal Geometry, and this has given rise to what is known as Geometric Measure Theory. Our goal, through this reading project, is to develop a basic understanding into this field. For the same, we shall follow two books:
Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer.
Geometry of Sets and Measures in Euclidean spaces: Fractals and Rectifiability by Pertti Matilla.
Through the reading project, we expect to make notes and lectures on the topic for future use.
Prerequisites: Multivariable Calculus, Linear Algebra, Metric Spaces, Measure Theory and Integration and basic Functional Analysis and Topology.
NOTE: No renumeration or certificates can be provided since the project is not funded by any organization. This project is purely based on my personal curiosity and willingness to learn the subject. Interested people can contact me through mail: aniruddha480@gmail.com.