# Research

## Fractal Dimensions Article

If you want an undergraduate level explanation of one of the topics studied in fractal geometry, check out the article I wrote on Fractal Dimension. I often point my students to this article when they ask what I study.

## Analysis on Fractals

Ever wonder how one can do calculus on the Sierpinski gasket? How can one define a derivative or even a Laplacian on a fractal set? And once one has a Laplacian, can we study differential equations which may tell us how heat dissipates on metals with fractal molecular structure? Or how water may move on a dish with a fractal boundary?

## Math for Social Justice

Through the RUMBA program, I lead three undergraduate research groups on how mathematics can be used to study the social, cultural, and political systems in the Bay Area. Check out the RUMBA website for more details.

## Noncommutative Fractal Geometry

Noncommutative fractal geometry is the study of fractal geometry and analysis on fractals, with operator algebraic tools. My work in this area focuses on how one can use spectral triples (C* algebra, a Hilbert space, and a Dirac operators) to formulate definitions of classic fractal geometric ideas such as dimension, metric, and measure.

## Data Science and Fractal Geometry

I have a group of graduate students currently exploring how one can use fractal dimension to quantify how much chaos exists in a data set. Data sets arrising from EEG machines are partiicularly interesting for this purpose. Known notions for assigning a fractal dimension to such data include Higuchi's dimension and Katz dimension.

# Arauza Research Group Members

### Undergraduates

Sandra Torres

### Graduate Students

Matthew McClinton

### Past Students

David Smith; M.S. 2021

Asees Kaur; M.S. 2020; now starting the Applied Math PhD program at UC Merced

Adam Lankford; B.S. 2021

Edwin Lin; B.S 2020; now starting the Math PhD program at UC Riverside