The pandemic summer of 2020 found me working with the Sierpinski gasket (SG) and its "spectral appearance". The information I was in looking for did not seem to be available first hand and I realized that there are still some gaps in our understanding of its structure. Out of the wish to delve into the spectral picture (eigenvalues and eigenfunctions) of SG the Undergraduate Fractals Research Team was launched in Fall 2020. Currently it is partially supported by an NSF grant.
During the academic year 2020-2021, the founding members Amy, Jessica, Angelica and Xochitl started coding an approximating fractal calculator and investigating several striking spectral features of SG. Their findings have laid the groundwork for a series of exciting projects ahead!
In this academic year 2021-2022, Jules, Catherine, Veronica and Emily continue these investigations and we look forward to posting you about new developments!
The spectrum of the Sierpinski gasket was described by Fukushima and Shima in a genealogical way that one can implement in a computer program. Later on, Kigami, Strichartz and many other authors made also possible to provide algorithms to construct eigenfunctions.
As a music chord is composed of different notes played unison, most signals that reach us - sounds, biomolecules, electromagnetic waves or stock prices - may be expressed as the composition of more basic components. These components can be classified by different frequencies and the more complex the signal, the more frequencies are involved in its composition. At the same time, we can build signals by adding as many elements from different frequencies as we wish.
In mathematical terms, signals are functions. Functions can also be decomposed into "more elementary" ones classified by their frequencies. A widely used type of "more elementary" are so-called eigenfunctions, which build what is classically called a Fourier basis. That basis provides what we could call the "spectral appearance" of the input. Because frequencies enjoy a basic structure, they can be quite useful, for instance to solve equations whose initial data can be described in terms of these frequencies.
While the classical theory of Fourier series can be applied to study many problems, it usually covers situations where the phenomenon subject of investigation - heat spread, wave propagation etc. - occurs in a rather “smooth” medium. If we are interested in solving these equations inside a highly porous material like a sponge, that standard Fourier basis is not available. One needs to work with a "spectral appearance" that fits the model.