New Method Extends Harmonic Function Approximation to Complex Surfaces

Bernstein‑Walsh harmonic approximation theorem now applies to Riemann surfaces, rate for approximating harmonic functions. Preprint: 9 Oct 2024, updated 26 Sep 2025. getnews.me/new-method-extends-harmo... #riemannsurfaces #harmonicapproximation

Orbits of Theta Characteristics The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for advancing the understanding of these orbits, generalizing existing results of Kallel & Sjerve, allowing us to establish the existence of infinitely many curves possessing a unique invariant characteristic as well as determine the number of invariant characteristics for all Hurwitz curves with simple automorphism group. In addition, we compute orbit decompositions for a substantial number of curves with genus <= 9, allowing the identification of where current theoretical understanding falls short and the potential applications of machine learning techniques.

"Orbits of Theta Characteristics" by H. W. Braden and Linden Disney-Hogg. #ExperimentalMath #RiemannSurfaces #MathSky

Analytic continuation on the complex plane sure looks a lot like selecting an atlas of charts on a complex manifold. #riemannsurfaces