Portrait of Sam Silverman '23. Photo by Patrick Montero

What They Learned: Samantha Silverman ’23

Wildfires, flooding, and differential equations are at the heart of Sam Silverman’s math thesis.

Sam Silverman, a double major in math and computer science, made the most of her time at Haverford College and hopes to make a difference in life after graduation. A native of Catonsville, Maryland, Silverman not only completed a double major, but also flourished as a varsity athlete on the women’s lacrosse team, earning both academic and athletics honors.

A double major brings two senior theses. For Silverman, math was actually the second major that she picked up even though it was foundational to her computer science major (for which she wrote a purely comp-sci thesis). Her math thesis allowed her to combine both math and computer science into one piece, and involved implementing the approximation method for the (partial differential equation) PDE into Python in connection with her interest in post-wildfire landslides.

She says that her math thesis advisor Timur Akhunov let her take control of her project. Though he got her started with an initial equation, she did all the research, found all the articles she used as background, and conducted the analysis herself.

“I had never taken a partial differential equation (PDE) course before, so I was a little hesitant to go down that direction, but he encouraged me to go this route, which allowed me to explore my topic in its entirety,” Silverman said. “Once I switched from ordinary differential equations (ODEs) to partial differential equations (PDEs), he found some textbooks and chapters to get me started in this learning, while answering my questions when they arose. He also reviewed and guided my writing throughout the process. Overall, my advisor was there when I had questions, but he really emphasized the idea of it being my project, which allowed me to steer it in the direction I wanted.”

Her math research allowed her to investigate landslides following wildfires, where she first analyzed post-fire landslides and flooding with partial differential equations (PDEs), using an approximation technique and implementation to identify the depth of water around a two-dimensional grid, short-term and long-term. She observed in most cases that the flooding reaches its peak within three minutes of the rain starting. She then supported this landslide analysis with the investigation of fires using ordinary differential equations (ODEs), specifically a single-ODE logistic model and a more involved system of two ODEs. She investigated what happens to trees and grass and what thresholds are necessary for them to survive in a predator-prey model. She found that when the grass grows faster than the fire spreads, the grass will survive.

“The relationship between math and the real world is so interesting to me, so I wanted a topic that allowed me to explore the connection between something current and relevant to the real world as well as use a form of modeling to do so,” said Silverman. “Connecting wildfires, flooding, and differential equations created a project that allowed me to analyze results with graphs and equations as well as gain knowledge outside of the classroom curriculum.”

 Silverman plans to work for the federal government in public service as an applied research mathematician. Although she doesn’t foresee interacting with this research directly, at least in the immediate future, she found it extremely interesting and the process was useful in that it can be applied to almost any problem set.

Silverman sees profound implications for her research. “Wildfires are happening often right now in California and worldwide, and once the land endures this kind of impact, it cannot always handle the presence of rain and erosion, which leads to the flooding that I investigated. I heard that scientists are even using twitter scraping to see when these floods are happening and modeling is often a good way to bridge gaps in sensing. Thus, modeling and trying to predict where and when the flooding will occur after a fire as well as how much flooding is bound to occur is something that could help protect the land and the surrounding neighborhoods and inform those that might be impacted. There is still a lot of research and modeling that can be done to help make this even more realistic, which would continue to help those impacted by the fires and floods. There could be improvements to the model I investigated by increasing the complexity and analyzing the water flow velocity in addition to the water depth. Also, analyzing the effects after the precipitation stops would help researchers better understand how long it would take for the flooding to recede after the rain stops. Overall, my thesis presents just a start in what could be something researched further in terms of understanding and predicting flooding and debris flow after wildfires.”

Ultimately, Silverman identified two major takeaways from the project. The first is learning about PDEs, including how to model them numerically and get results in a form that she could interpret using graphs. while The other involves exploring the connections between the actual partial differential equation she used and the approximation method that she implemented in Python and graphed. “They ended up matching up almost entirely,” she says, “which was helpful in my learning process. Something else I learned was that this type of writing is not linear, and many edits can be made, and sections added, but sometimes earlier versions and fewer words are better. Another takeaway I had is that the results seemed realistic, which is so interesting to me that I could take a bunch of parameters and a model and graph something that could be extended to real life.”

 “What They Learned” is a blog series exploring the thesis work of recent graduates.