Persistent Homology and Computer Vision
One of my latest projects uses ideas from discrete morse theory and persistent homology to identify defect patterns in Rayleigh-Bénard convection. The following video was made from simulated data from the Paul Research Group at Virginia Tech (experimental data taken at the Mike Schatz laboratory). Highlighted regions in the persistence diagrams indicate parameters for pattern-matching algorithms based on the output of an application I’ve been working on.
TDA Persistence Explorer
Ever wonder how the points in your persistence diagram correspond to the features in the digital images you’re processing? Download the tda-persistence-explorer app from GitHub and explore your data! Based on the persistence computations from PHAT that utilize discrete Morse theory, this software enables you to encircle persistence points of interest and will then display the corresponding critical cell pairings overlaid on your image, as the screen shot below shows. It also includes a reverse search feature: select a region of interest on the image and the tool will highlight any persistence points with critical cells from their underlying pairings that fall inside.
The tool is also great for studying time series of persistence diagrams generated from digital images. The software includes an installation script that will locally install a copy of PHAT and other required dependencies. It also includes a Jupyter notebook that will get you started on some test data. The test images are numerical simulations of Rayleigh-Benard convection flow and were generously provided by the Paul Research Group at Virginia Tech.
I was involved in a research project concerning inequalities of singular values of positive semi-definite matrices under the direction of Dr. Raluca Dumitru at the University of North Florida during the Fall 2011 term. We were able to reduce the inequality in question and our results were published in Linear Algebra and its Applications.
PRESENTATION: Imagining the Banach-Tarski Paradox. Presented at the Pi Mu Epsilon National Meeting at MathFest 2011, in Lexington, KY, on August 4, 2011 and received a student speaker award.
ABSTRACT: The Banach-Tarski Paradox states that it is possible to take a solid ball in three-dimensional space, divide it into a finite number of non-overlapping subsets, and then recombine those sets to create two balls with the original size and volume of the first, thus duplicating the sphere. In this presentation, we will examine the precise formulation of duplicative paradoxes, and then follow the development of the Banach-Tarski Paradox from the hollow sphere to the solid ball. With the use of Mathematica pictures and animations throughout the presentation, we will provide a visual interpretation of how such a decomposition is possible, aiding in developing an intuition for the sets involved.
An expository writing concerning the development of the Banach-Tarski Paradox, aimed at the advanced undergraduate level. Served as an introduction to LaTeX and Mathematica. Work performed under the direction of Dr. Scott Hochwald at the University of North Florida, Spring 2011.
Miscellaneous Mathematical Interests
Mathematics as Metaphor: Mathematics in Contemporary Art. A paper I wrote for a course in Contemporary Art, Fall 2010, under the instruction of Dr. Elizabeth Heuer at the University of North Florida.
ABSTRACT: By examining the relationship between mathematics and art from a historical standpoint, and then looking at the works of Bruce Nauman and specifically his works that use the mathematical subject of topology as motivation, we will see a way in which mathematics may be used to inform contemporary art by way of artistic process and not merely through illustration.
Dueling Scribbles. A small JAVA program that randomly generates two “scribbles” according to a mathematical set of rules (press “d” repeatedly to redraw the scribbles), and the beginnings of an art project on computer generated art and identity. What surprised me in this project was that the nature of the scribbles, though random, seemed to betray the non-randomness of the random number generator used to create them. When drawn, sometimes both will be clumped up in a corner, or both will be splayed out across the page, even though each scribble is instantiated entirely separately from the other.
A Prayer for Math, 2011 (Fib Poem)
language of nature,
help me see the world through new eyes.
I am currently a postdoctoral researcher at the University of Pennsylvania in the Department of Electrical and Systems Engineering, where I work with Professor Robert Ghrist on both theoretical and applied aspects of persistent homology. I received my Ph.D. in mathematics at Rutgers University in May 2017 under the direction of Professor Konstantin Mischaikow. Please download my CV and research statement. Here are some papers on arXiv.org that I had a part in writing. I maintain my active projects on github.