Abstract: 

This mini-course is based on a joint work with Jason Miller [2], where we obtain results on all geodesics in the Brownian map, including those between exceptional points. 

I will start by presenting 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. I will give the main ideas of its proof, and in particular present a useful tool which is the breadth-first exploration of the Brownian map. 

I will then present results on the classification of the topology of geodesics between any pair of points. Among other things, 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. 

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.  

It is helpful to have some knowledge on the Brownian map, e.g., to have read [1], but it is not an obligation and I will try to make this course self-contained.

[1] Le Gall and Miermont. Scaling limits of random trees and planar Maps. Clay Mathematics Proceedings. https://www.imo.universite-paris-saclay.fr/~jflegall/Cours-Buziosf.pdf

[2] Jason Miller and Wei Qian. Geodesics in the Brownian map: strong confluence and geometric structure. https://arxiv.org/abs/2008.02242

Time: 
 1) September 8, 4-5 pm in Beijing time,
 2) September 9, 4-5 pm in Beijing time,
 3) September 10, 4-5 pm in Beijing time.

Zoom: 

https://us02web.zoom.us/j/87095704312?pwd=SytOcFVVN1dnNkxvOUFDWnRFVGFJUT09
ID：870 9570 4312
Code：880256