TDA Research Group

At my current position with the Science Research Initiative at the University of Utah, I lead a group of undergraduate researchers in the area of Topological Data Analysis.

Students in my research group work on a variety of projects in both theory and applications, including:

Enumerating and visualizing the fibered barcode for bi-persistence modules,
Computing and understanding filtrations and persistence modules built on a fixed set of vertices,
Using TDA to understand neural data, genetic mutations, and physics, and
Creating a educational video series on persistent homology.

Stream Skills

Developing proficiency in reading and writing mathematics
Developing qualitative technical communication skills
Quantifying and modeling questions from all areas of science with mathematics
Proof writing
Creating algorithms, and implementing them with coding

Who should apply for this stream?

Students who want to find creative ways to answer quantitative questions in a wide range of scientific fields!
Students who want to develop their technical communication and qualitative reasoning skills!
No prior math experience required!

Research group projects

Applications of TDA to Physics

Topological Data Analysis (TDA), an emerging data science field, can be used to uncover patterns in different areas of physics, such as the structure and behavior of fusion plasma. For fission, it can be used for patterns in nuclei splitting. By applying TDA we can find hidden features in the data (clusters, loops, or holes in the data) that may help us find physical phenomena such as the unexplained losses in a fusion reaction or the unique way heavy atomic nuclei move. Ultimately our goal is to understated how different areas of physics may benefit from using TDA.

Accessibility to Primary Healthcare

Taking into account 4 common parameters to obtaining primary/preventive care (income [including insurance coverage], language, wait time, and distance), I am attempting to quantify the difficulty level for going to visit a provider. This is specific to Salt Lake County, but the methods being developed will be able to be used for any desired location. TDA will be used to analyze the geographical data as well as demographic data.

Vietoris & Rips: an educational TDA video series

Vietoris and Rips are puppet characters that are here to help anyone, even the youth, learn about Topological Data Analysis! With specialized episodes, anyone can learn about TDA topics like simplicial complexes, dimensionality, filtrations, real-world applications, and more. Our goal is to make these videos accessible online and in classrooms, with workbooks that go along with the videos, to create an engaging and fun learning environment.

Enumerating equivalence classes of the fibered barcode of bi-persistence modules

In topological data analysis, the persistent homology pipeline allows us to understand topological information about a data set in the form of a barcode – a summary of persistent homological features (“holes”) across a one-dimensional filtration of simplicial complexes on our data. Some data warrant investigation across multiple dimensions, requiring the use of filtrations indexed by more than one parameter. Key information about these multidimensional filtrations cannot be completely described by one barcode; instead, summarizing the data requires a collection of barcodes, where each barcode corresponds to a one-dimensional filtration of our data. Critical points, points in the filtration where the simplicial complex undergoes a change in homology, partition such one-dimensional filtrations into equivalence classes by barcode. Our research aims to enumerate and describe these equivalence classes. Equivalently, in two dimensions, we want to understand how many ways a line with positive slope can partition a set of points in the plane.

Modeling Horizontal Gene Transfer with TDA

My research is interested in the biological application of topological data analysis and what it can tell us about lateral gene transfer of antibiotic resistance in populations of bacteria. The relationship of resistance encoding genes on bacterial plasmids is a key component to how antibiotic resistance arises in bacterial pathogens through horizontal gene transfer. TDA is a useful tool which can reveal when and how often these reticulate events occur given the genetic distances of bacteria, their populations, and between a population over multiple generations. My project aims to use RStudio and TDA to model and analyze how often reticulate events occur for three distinct bacterial species and their populations undergoing transduction, transformation, and conjugation.

TDA to Model Brain Degradation Under Sleep Deprivation

Sleep deprivation is an ever-increasing issue in our fast-paced modern world, and while we all know how hard it is to operate after a bad night of sleep, the longer-lasting effects of sleep deprivation aren’t well known. Through Topological Data Analysis, we want to try to address this issue. We analyze the structures of the brains of people who have undergone significant sleep deprivation, through synthetic data sets, to see what parts of the structure have changed from a healthy brain. Initially, we plan to do this with small regions of the brain, such as the hippocampus, before putting our research together to provide a more complete view of our brains after periods of sleep deprivation.

Topological Measures of Psychoplastigen Induced Neural Plasticity

This project focuses on using topological data analysis (TDA) to measure changes in dendritic arbor complexity, with the aim of creating new metrics that go beyond traditional tools like Sholl analysis. Looking at neurons through topology preserves more of the information about how their complexity shifts, which could reveal differences in how those changes unfold with the use of different plasticity-inducing substances over time. Building a topology-based tool to capture the qualities of dendritic growth has the potential to deepen our understanding of neuroplasticity in ways that matter across many areas of neuroscience.

Enumerating unlabeled and labeled simplicial filtrations on N vertices

Our research aim is to study the ways in which a clique complex filtration on fixed n vertices can be constructed, with an eye for filtrations producing equivalent barcodes and/or certain topological features. Further we eek to explore and validate such notions of "equivalence," for clique complexes, simple graphs, filtrations and/or barcodes. This research is motivated by the growing need to understand information about filtrations on sparse data.

Research group members

Eve Bradley, Research Assistant

Bio: My name is Eve Bradley, and I’m an undergraduate studying math and electrical engineering at the University of Utah. I’ve been researching topological data analysis with this stream since January 2024, and I love being part of such an encouraging math community! I’m also involved at the U as a tutor at the math center, an ambassador for the College of Science, and a TA and peer mentor for the ACCESS Scholars program.

When I’m not running around campus, I like to take walks with my older sister, watch movies with my roommates, draw pictures, listen to music, play the ukulele, and do all sorts of puzzles.

Matthew Johnson, Research Assistant

Bio:

Evan Birkinshaw, Research Assistant

Bio: Hello! My name is Evan Birkinshaw, and I am an undergraduate student at the University of Utah pursuing a double major in bioinformatics and cellular and molecular biology. I am a member of the Science Research Initiative and have been a part of the TDA research stream since spring of 2024 and as a research assistant as of summer 2025.

My area of research within the stream is focused on the biological application of TDA with respect to viral topology interested in what the topological structures of horizontal gene transfer in bacterial plasmids can tell us about antibiotic resistance and bacterial evolution.

Apart from my research, I am an athlete on the University of Utah’s rock-climbing team, I enjoy outdoor recreation- especially backpacking, reading classic novels and nonfiction, listening to obscure alt music, writing, cool clothing, and hanging out with my awesome friends and TDA stream!

Debbie Wooton, Research Assistant

Bio: Hi! I'm Debbie Wooton, and I graduated from the University of Utah with a bachelor's degree in mathematics in May 2025.

Since Fall 2022, I've participated in the Science Research Initiative, initially studying higher-dimensional chip-firing, then learning about Barile-Macchia resolutions of matroid ideals, and now researching fibered barcodes of bipersistence modules.

In my free time, I like to read , and play video games, board games, and Dungeons & Dragons!

Josie Marshall, Research Assistant

Bio: My name is Josie Marshall and I am from Park City, Utah. In December 2024 I graduated with my bachelor's in applied mathematics and I am currently studying at the University of Utah in the Master of Information Systems program with an interest in ERP Systems Management and Predictive Analytics.

My experience with the Science Research Initiative includes topics in Topological Data Analysis, High-dimensional Chip-Firing systems, and Barile-Macchia Resolutions of Matroid Ideals.

Outside of school I love to golf, ski, and try new recipes!

Henri de St. Germain

Bio: Hi! My name is Henri de St. Germain and I’m currently a sophomore studying mathematics at the University of Utah. I was born in Salt Lake City and have lived here my entire life. I have been a member of the stream since spring 2025. In my free time I like to hike, read, ski, and play Ultimate frisbee.

Apsen Warden

Bio: Hi! My name is Aspen Warden, and I’m pursuing honors degrees in Bioinformatics and The Philosophy of Science at the University of Utah. I am an ACCESS Scholars alumna, and alongside my work in Dr. Brooks’ lab I volunteer with the HMHI U-PSI lab as a research assistant and IRB member, and with The Zendo Project as a logistics and log keeper coordinator. I also work as a College of Science ambassador with the University of Utah. Alongside school and research, I love live theatre, playing guitar, yoga, spending time with my family (especially my dogs) and creating and consuming art!

I’m interested in how psychoplastogens drive structural neural plasticity, and in using topological analysis to develop new ways of measuring changes in dendritic arbor complexity.

Oliver Duncan

Bio: My name is Oliver Duncan and I am originally from Pleasant Grove, Utah. I am presently pursuing my bachelor's degree in Mathematics at the University of Utah, with expectations to graduate in 2028.

My time with the Science Research Initiative has been primarily occupied with Topological Data Analysis, with slight emphasis given to pursuits in multi-parameter persistence.

When left unattended, I have been known to enjoy reading, calligraphy, cinema, music, and games of all shapes, sizes, and methods of ingestion.

Benjamin Terry

Bio: My name is Benjamin Terry. I live between Millcreek and Sugar House with Oliver. I am a Chemical Engineer major, and I am in my sophomore year. I like to mountain bike and play video games. I like to play the guitar and ukulele. This stream group is awesome sauce.

Ryan Friel

Bio: Hello! My name is Ryan and I’m from Taylorsville, Utah. I am currently studying Math at the U with a minor in Chemistry. I am finishing up my 2nd year and plan to apply for medical school in the coming years. I love to fix/ride bikes, play soccer, and practice piano when I have some time. My project in the TDA stream focuses on accessibility to primary healthcare within Salt Lake County. I hope to quantify the difficulty some demographic and socioeconomic groups have while searching out professional medical help.

Tori Roper

Bio: Hello! My name is Tori Roper and I am a Math Education major here at the U and minoring in Spanish. As an alumnus of the ACCESS Scholars program, I'm part of the Science Research Initiative doing work in Topological Data Analysis. My project in this lab specializes in creating educational videos geared towards younger audiences, as well as anyone looking to learn, in order to popularize and inform about this lesser known, but incredibly useful form of math. Outside of school, I love hanging out with friends, playing the piano, and playing tennis.

Montader Alasady

Bio: My name is Montader Alasady and I am from Murray, Utah. I expect to graduate with my bachelor’s in Computer Science in 2027.

My experience with the Scientific Research Initiative is centered around Topological Data Analysis, specifically towards applications in Physics.

My hobbies include reading fiction, playing games, and watching movies.

Addis Challenger

Bio: I’m Addis Challenger, a Sophomore mathematics major from Lehi, UT.

My interests are currently found in using TDA to model the brain after potential instances of degradation, such as sleep deprivation. Albeit, I’m interested in a wide variety of mathematics problems, including the work of a lot of other members in this stream.

I’m a big fan of grand strategy games, mainly Hearts of Iron IV and Victoria III, although I’m not very good at them.

Dawson Wheeler

Bio: I plan to use TDA to understand the shape and underlying structure of biological data related to neurodegenerative diseases like Alzheimer's. To do this I’m going to analyze different datasets such as neuroimaging, omics, and physiological data, which can hopefully shine light on subtle topological signatures that are overlooked by traditional methods. This approach could potentially aid in the early detection of disease and the identification of new therapeutic targets, such as shared glial stress pathways across different conditions.

Ali Al-Barkawi

Bio: My name is Ali Al-Barkawi. I am currently attending the University of Utah as a Computer Engineering major. My interests are embedded systems and PCB design.

Our group will be doing Topological Data Analysis on sustained nuclear reactions in a contained system. I.E., nuclear fission or fusion. Our main goal is to see the patterns in sustained nuclear fission reactions and plot a topological map. We also might move our research toward fusion reactions and see if we can find where fusion reactions struggle to sustain their reaction.

Hank Dolan

Bio: My name is Hank Dolan. I’m a sophomore math major, and I’m originally from Ogden, Utah. I plan to graduate in Spring of 2028, and before that, I’d like to do as much interesting math as possible, and this stream has provided me with many interesting mathematical opportunities!

I’m primarily interested in theoretical mathematics, and this stream has allowed me to explore that interest. It has also given me a mathematical community that has helped my mathematical communication skills and allowed me to meet new people.

Outside of school, I really love music and play the banjo and drums. They’re an awesome creative outlet, and they’re sometimes also weirdly mathematical.

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