I will talk about our recent results on all geodesics in the Brownian map, including those between exceptional points. This is based on joint work with Jason Miller (https://arxiv.org/abs/2008.02242).
First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints.
Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each k=1,…,9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7,8,9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case.
Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.