The pandemic summer of 2020 found me working with the Sierpinski
gasket (SG) and its "spectral appearance"1. 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.
Current members
William Echols
Oluwaseyi Johnson
Lois Ruiz
Savannah Young
Luz Acevedo, AY 22-23
Mayra Hernandez, AY 22-23
Lucy Zhang, AY 22-23
Julianna Cruz, AY 21-22
Emily Vasquez, AY 21-22
Veronica Flores, Fall 2021
Chau Duong, Fall 2021
Angelica Benitez, AY 20-21
Xochitl Maldonado, AY 20-21
Jessica Nunez, AY 20-21
Amy Zhou, AY 20-21
With AJAX in Spring 2021
Emily & Jules in Spring 2022
Luz, Mayra and Lucy in Summer 2022
During the academic year 2020-2021, the founding members Amy, Jessica, Angelica and Xochitl
coded and investigated several striking spectral features of SG. Their findings laid the groundwork
for a series of exciting projects ahead.
The new 2023 team will move forward our investigations and we look forward to posting you about new developments!
More about the topic
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.
