Research Summary

My research interests include topological data analysis with applications to dynamical systems and biological phenomena. I am also interested in the intersection of topological statistics and machine learning. In particular, I am interested in answering questions related to the robustness of topological summaries in noisy systems, the predictive power of topological statistics in modeling, and the effectiveness of topological quantities as input for machine learning tasks. At present, I work in a range of applications including zebrafish stripe development and evolutionary biology. Below I give a very brief description of my research projects. Feel free to contact me for more information.

My research is currently being funded by the National Science Foundation Graduate Research Fellowship Program. Any opinions, findings, and conclusions or recommendations expressed on this website are my own and do not necessarily reflect the views of the National Science Foundation.

Current Research Projects

A Topological Analysis of Model Sensitivity and Classification for Pattern Formation on Zebrafish
My dissertation research is advised by Björn Sandstede and co-advised by Andrew Blumberg. We are interested in applying topological data analysis to study dynamical systems and biological phenomena. Currently I am working on a problem related to classifying model outputs from zebrafish stripe development models. We use topological summaries of the model outputs as input to a classification algorithm. This gives us a way to automatically classify model outputs under various parameter regimes and noise tests. We hope to use this work to better understand these models and the underlying biological mechanisms.

Topological Estimation of Recombination Rates
I collaborate with Devon Humphreys and Michael Miyagi under the supervision of Andrew Blumberg on this project. Briefly, we extract topological summary statistics from genomic data and then use techniques from regression analysis to infer hotspots of recombination.

Research with Undergraduates
In the Summer@ICERM 2017 program I began working with Dr. Katherine Kinnaird and undergraduates Erin Bugbee, Claire Savard, and Jonathan Weisskoff on the cover song task. The goal for this project is to develop a flexible and computationally efficient method for completing the cover song task. We use methods inspired from topological data analysis to correctly match songs which are remakes of the same original piece. This work in ongoing with Katherine Kinnarid, Erin Bugbee, and Claire Savard. Congratulations to Claire and Erin for winning the MAA "Outstanding Poster Award" at the 2018 Joint Mathematics Meeting where they presented our work!

Research Group

Professor Sandstede's research group participates in weekly meetings each semester where we discuss topics in dynamical systems and related fields. Below is an overview of our meetings along with the presentations I have presented.

Semester Main Theme My Subtopic Presentation
Spring 2018 Probability and Statistics Classification Algorithms Lecture and Python Demo
Fall 2017 Dynamics and Statistics Parallel Computing in Matlab Interactive demos
Spring 2017 Data Science Methods of Machine learning Here.
Fall 2016 Vegetation Patterns TDA and Diffusion Maps Here.

Past Research Projects

A Topological Analysis of Targeted In-111 Uptake in SPECT Images of Murine Tumors
I worked under the supervision of David Damiano for my undergraduate thesis at The College of the Holy Cross. We developed a novel topological method of analyzing Targeted In-111 Uptake in SPECT Images of Mouse Tumors. The motivation for this method is illustrated in the videos below. The videos below show the super levelsets of images of murine tumors as we descend through a filtration of tumor uptake values. The video on the left corresponds to the mouse tumor at hour 24 in the study, and the video on the right is the same tumor at hour 72. Our method captures the topological information of these filtrations of the tumors across a time series of images and surpasses standard techniques in their ability to capture tumoar heterogenity. You can find our paper here.

Assimilating Eulerian and Lagrangian data in traffic-flow models
During the summer of 2014 I participated in a research project on data assimilation for traffic flow through a Research Training Grant on “Integrating Dynamics and Stochastics.” I worked under the supervision of Björn Sandstede and collaborated with Courtney Cochrane, Joey DeGuire, Bridget Fan, Emma Holmes, Patrick Murphy, and Jenna Palmer. Our graduate mentors were Paul Carter, Laura Slivinsky, and Chao Xia. You can find our paper here.