Research / Talks

The Oscillatory Behavior of Relative Fractal Drums Associated with a Class of Space-Filling Curves

This talk discusses the how the relative fractal drum (RFD) I constructed can detect the oscillatory behavior associated with a class of space-filling curves, in particular the Hilbert curve and other curves like it. The oscillatory behavior is present in the 1-dimensional components of the curves (the points) and in the lower order terms of the volume formula for the epsilon-tubular neighborhood of the curve. I used Mathematica to generate a 3D model of (two of) the RFDs I designed, images of which can be seen as the background on my homepage and the Github link page. I am curious about what applications this construction could be used for, and wonder if it could be used in image- or data-processing since it is associated with space-filling curves. I gave this talk at the Mathematical Physics and Dynamical Systems (MPDS) seminar at UCR.

A Generalized Relative Fractal Drum for Plane-Filling Curves

This is the talk I gave for my Oral Examination at UCR and it details the dissertation result I found a few months earlier. More specifically, I constructed a relative fractal drum (RFD) capable of detecting the complex dimensions of the Hilbert curve, Peano curve, and any plane-filling curve constructed in an analogous via a regular rectilinear tessellation. This is exciting because it is the first time these curves can be properly classified as fractals using the most accurate definition of fractality, introduced and developed by my advisor Dr. Michel L. Lapidus over the past 30 years. I have also given this talk in the Fractal Research Group (FRG) seminar at UCR, MPDS seminar, and the AMS Sectional Meetings.

Finitely Additive Invariant Set Functions and Paradoxical Decompositions, or: How I Learned to Stop Worrying and Love the Axiom of Choice

This is a talk I gave at the CSULB Mathematics Colloquium in 2018 and the AMS Graduate Student Seminar at UCR in 2019 about the Banach-Tarski Paradox and how it works. The paradox certifies that no finitely additive invariant set function (finitely additive measure) can be established for all sets in Euclidean space when the dimension is 3 or greater. In other words, no intuitive definition of measure can be established for all sets in R^3 or above, only for a very large class of sets known as measurable sets. The Axiom of Choice guarantees such paradoxes, but eliminating it as an axiom eliminates many fundamental results of analysis.