Cheenta School Research - Hurwitz Investigations

Jayaditya Gupta, a Class 10 student from Mumbai, India, has prepared an inspiring presentation on Euler’s Characteristic and Herz’s Investigations in Group Theory. Jayaditya was part of the Hyperbolic Geometry Group during the summer program, and his talk beautifully bridged the worlds of geometry and algebra.

In her presentation, Jayaditya began with the elegant Euler characteristic formula for surfaces of different genus, guiding the audience through detailed proofs using intuitive visual models like the double torus. From there, he transitioned into the fascinating world of group diagrams, using examples like the symmetric group S3S_3S3​ to demonstrate how group relations are represented geometrically.

The highlight of his talk was the detailed discussion on Herz’s investigations, where he examined how group diagrams evolve on hyperbolic surfaces and how the number of group elements relates to the genus of the surface. He concluded by connecting these ideas to the word problem in group theory, emphasizing how geometric tools can help solve abstract algebraic challenges.