Study Paths, Riemann Surfaces And Strebel Differentials


  • Peter Buser École Polytechnique Fédérale de Lausanne
  • Klaus-Dieter Semmler École Polytechnique Fédérale de Lausanne



Study Paths, Riemann surfaces, Conformal Geometry


These pages aim to explain and interpret why the late Mika Seppälä, a conformal geometer, proposed to model student study behaviour using concepts from conformal geometry, such as Riemann surfaces and Strebel differentials. Over many years Mika Seppälä taught online calculus courses to students at Florida State University in the United States, as well as students at the University of Helsinki in Finland. Based on the click log data of his students in both populations, he monitored this course using edge-decorated graphs, which he gradually improved over the years. To enhance this representation even further, he suggested using tools and geometric intuition from Riemann surface theory. He also was inspired by the much-envied Finnish school system. Bringing these two sources of inspiration together resulted in a promising new representation model for course monitoring. Even though the authors have not been directly involved in Mika Seppälä’s courses, being conformal geometers themselves, they attempt to shed some light on his proposed approach.


Hancock, L. (2011). Why are Finland’s schools successful? Retrieved from: http://www.

Seppälä, M. (2013). Learning analytics, Riemann surfaces, and quadratic differentials, in: Rie- mann and Klein surfaces, automorhisms, symmetries and moduli spaces, Link ̈oping, Sweden, June 24-28, 2013. Prerecorded in: Cum07xg

Seppälä, M. (2014a). Shape of educational data, Florida State University, Tallahassee (FL), September 2, 2014. Recorded in: k

Seppälä, M. (2014b). Geometry of data, Florida State University, Tallahassee (FL), September 23, 2014. Recorded in:

Strebel, K. (1984). Quadratic Differentials. Ergebnisse der Mathematik und Ihrer Grenzgebiete. Springer-Verlag, Berlin 1984.

Ecole Polytechnique Fe ́de ́rale de Lausanne, FSB–Mathematics Institute of Geometry and Applications– MATHGEOM, Station 8, CH–1015 Lausanne E-mail address:




How to Cite

Buser, P., & Semmler, K.-D. (2017). Study Paths, Riemann Surfaces And Strebel Differentials. Journal of Learning Analytics, 4(2), 62–75.



Special section: Shape of Educational Data