Brin Postdoctoral Fellow Hannah Hoganson’s love of mathematics inspired her to study geometric group theory and to rethink the way math can be taught.
Hannah Hoganson’s passion for math took her across the U.S.—from her hometown in Pennsylvania to Ohio to Utah—before she finally arrived at the University of Maryland as a postdoc in fall 2022.
“Even though I always liked math and was pretty good at it, I wasn’t always in the highest-level classes as a kid. In fact, I even struggled with a couple math classes in college,” Hoganson recalled. “But after I attended more classes and experienced more opportunities, I realized that sometimes, you just need more time and more exposure to content to really understand it. Having different perspectives or different teachers can make difficult things finally click.”
Hoganson brought this philosophy with her as she conducts research and teaches courses at UMD as a National Science Foundation Postdoctoral Fellow and one of five Brin Postdoctoral Fellows who are supported by a generous gift from Mathematics Professor Emeritus Michael Brin. The Brin Postdoctoral program supports young mathematicians whose research shows remarkable promise for up to three years.
Hoganson works with Assistant Professor of Mathematics Lei Chen in the field of geometric group theory, a branch of math that involves using geometric tools and objects to study algebraic groups (such as a set of functions). By examining geometric symmetries or other behaviors like a geometric object’s possible transformations, Hoganson and Chen can learn more about the structure and properties of algebraic groups.
“Think back to the algebra classes you used to take—you have symbols and equations that you’re manipulating to find answers. It’s all fairly abstract,” Hoganson explained. “Now add in what you learned in geometry class, like measuring shapes and angles. Those are more physical interpretations or visualizations of the abstract. I’m interested in finding connections between the two, kind of like creating a ‘dictionary’ that translates one concept by using the other.”
Similar to how she finds connections between two different concepts in her research, Hoganson has a knack for finding ways to connect her work to people as well. Her experience partnering with peers, leading discussions and organizing conferences with mathematicians from across the country helped her become an effective communicator. She thrives when she’s bouncing ideas off collaborators, often moving her research forward after considering different perspectives and possibilities as she takes on intricate problems.
“Even if we’re all in the same field of study, we don’t always have the same ideas, understanding of concepts or conclusions,” she explained. “We have to know how to explain our findings and theories with each other in a way that we can all understand. And that requires the ability to tweak what you’re saying to different people and communicate well.”
And Hoganson practices what she preaches. She works to improve her teaching every semester, adjusting her lessons to each new batch of students by relying on feedback and her own positive experiences with allowing students to revise and resubmit assignments. Chen, who is Hoganson’s mentor, believes that these efforts are the reason why Hoganson is an effective teacher—and that sentiment is shared by the many undergraduate students who have taken her introductory calculus classes at UMD.
“Hannah is a super enthusiastic and talented mathematician,” Chen said. “She likes to work with other people, often communicating with various mathematicians and organizing new research activities with them. This love for collaboration and communication is also why she is a fantastic teacher. I cannot wait to see her as a professor.”
The path to math
Hoganson’s journey to UMD began when she attended a summer research experience for undergraduates program held at Miami University.
“I didn’t have much experience that first summer there, but I thought it was cool how math research worked. It was complex and dynamic. As an undergrad, you don’t usually get the chance to see that,” said Hoganson, who was a math major at Lehigh University at the time. "The program left such a big impression on me that I went again—this time, as the graduate student mentor for the undergraduate participants. It ignited my interest in continuing this sort of work."
Inspired by her undergraduate research experiences, Hoganson completed a master’s degree in mathematics at Miami University and later earned a Ph.D. in mathematics at the University of Utah. She found her research calling in low dimensional topology (a branch of math that studies shapes and spaces that can be visualized in two or three dimensions) and geometric group theory. She also resolved to bring that same exhilarating experience she had as an undergrad to other young mathematicians.
As she neared the end of her doctoral studies, Hoganson wanted to explore new research opportunities, and that eventually led her to UMD.
“I was drawn to UMD because I wanted a chance to work with Lei Chen, whose work shares many overlaps with my own research interests,” Hoganson said. “Another big attraction was the Brin Mathematics Research Center, which was brand new when I first learned about it. Having a big research center dedicated to pure math research was something that really set UMD apart from other schools and I knew I wanted to be a part of that.”
Since landing the postdoctoral position at UMD, Hoganson has gone on to teach several introductory math classes at UMD and publish several papers on mapping class groups with longtime collaborators and friends. She’s taking a break from teaching this semester to focus more on her research but hopes to return with new teaching tactics and an updated approach to conveying new ideas.
“It’s definitely a good time to be here at UMD,” said Hoganson. “There’s a lot going on right now in terms of research, workshops and collaborations that I look forward to exploring.”
Written by Georgia Jiang