Auburn mathematics professor publishes book using methods developed during the pandemic

As Hal Schenck prepared to teach his applied topology course during the summer of 2020, the pandemic was in full swing, and faculty across the country were struggling with how to teach amid the challenges of hybrid learning and reach students despite a mainly digital presence.

Schenck never imagined that his summer class on applied topology and data analysis, taught under the shadow of COVID-19, would develop into a project that impacted students and researchers outside of Auburn. Based on the needs of his class – mainly students in statistics — and the constraints of the pandemic, Schenck modified both the teaching methods and course material to create a course that was unique in both format and content.

Teaching complex mathematics can be challenging even in optimal in-person settings. Schenck tackled the issue by developing new learning approaches that blended online lectures with collaborative problem-solving and office hours.

“What made the class a success was the willingness of the students to adapt to a new learning environment and the fact that I was able to pivot on the content to adapt it to meet the needs and interests of the students,” said Schenck.

As the students emanated from statistics, a need existed to fill in some of the pure mathematics background required for the material. Additionally, the field of applied topology continues to evolve rapidly, resulting in a lack of textbooks appropriate for the class. To address this, Schenck developed a set of course notes that built up the necessary algebraic background material from scratch while bringing students to the forefront of current research techniques.

“After the course ended, I realized that we had a set of notes that provided a unique approach to data analysis, so it made sense to polish it up into something useful to others,” Schenck said.

The resulting book, “Algebraic Foundations for Applied Topology and Data Analysis,” was published by Springer Verlag (the leading mathematics publisher) as the inaugural volume in their “Mathematics of Data” series. It comes as Schenck, the Rosemary Kopel Brown Chair in mathematics, completes a semester as a Leverhulme Visiting Professor at Oxford. The fellowship is funded by the Leverhulme Trust, a UK grant-making organization that advances research and education.

Aimed at practitioners from different research areas, the book begins with the fundamentals of the field, fills in the basics, and then builds gradually to advanced topics. Complementing the theory are numerous examples, ranging from a discussion of how Google’s page rank algorithm works to problems in robot motion and sensor networks and a discussion of how social media networks can be studied using graphs and the heat equation.

In addition to applied topology, Schenck’s teaching at Auburn has included courses in linear algebra, abstract algebra, calculus and computational algebra. He works on various topics besides applied topology, including physics and dynamics, approximation theory and geometric modeling and trading networks.