Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 20, 2025, 12:15 pm

Duration: 30 minutes

Location: OAT S15

Speaker: David Fisac (Universitat Autònoma de Barcelona)

A combinatorial perspective on Markov numbers (through geometry)

In this talk, we will introduce the well-known—yet still open—Markov Uniqueness Conjecture on Markov numbers, first proposed by Fröbenius in 1903. We will explore how the geometry of the modular torus leads to a reformulation of the conjecture in terms of the lengths of simple closed curves. Furthermore, we will show how the torus’s hyperbolic structure allows a purely combinatorial reformulation involving almost-equidistributed necklaces of positive integers. This new perspective has yet to be exploited for reproving known partial results or uncovering potential advances. The talk is intended for a broad audience, and no prior background will be assumed.

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