Directed by Dr. Kasso Okoudjou
In this project, students will learn the theory of calculus on fractal sets, namely the Sierpinski Gasket (
) and the Bubble-diamond fractals (
). We will learn how to develop some calculus notions on these fractal sets. The starting point is the construction of an operator, 
, called the Laplacian operator, which is analogous to the second derivative operator we learned in calculus. We will consider two classes of problems.
- We can then define the space of polynomial of degree less than or equal to
as the set of solutions of the equation 
. Over the last few years, students have worked on developing some orthogonal polynomials on these fractal sets. Our goal this summer will be to continue some of these constructions for various inner products. It is known that the space of polynomials on is not dense in the space of continuous functions on . We will explore similar questions for polynomials on . - One of the main properties of fractals is that they are self-similar sets. It is possible to endow the set on natural numbers
with a self-similar structure that comes with a natural Laplacian operator. In this project, students will explore several Laplacians on the nonnegative integer and investigate the spectrum of some Laplacian-based operators.
Desired background: Students interested in this project, must have completed a calculus sequence course, a linear algebra and/or a differential equation course. Having taken a proof-based course and a familiarity to a computer programming language, e.g., Matlab or Python, are also recommended.