Geodesic nets on Riemannian manifolds is a natural generalization of geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere.

In this talk I am going to survey some results and open questions (old

and new) about geodesic nets on Riemannian manifolds. Many of these

questions and results are about geodesic nets with edges of multiplicity one on the Euclidean plane. In particular, I will present a proof of a recent theorem by Fabian Parsch asserting that such a geodesic net with three boundary vertices can have at most one extra vertex of degree $>2$. I will also explain why no analogues of this result can hold for a large number of boundary vertices.

# Geodesic nets: examples and open problems.

Alexander Nabutovsky

University of Toronto; Member, School of Mathematics

February 26, 2019