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SUMMARY:Geodesics in the Brownian map: Strong confluence and geometric str
 ucture - Wei Qian (Orsay)
DTSTART:20210208T140000Z
DTEND:20210208T150000Z
UID:TALK157309@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:I will talk about our recent results on all geodesics in the B
 rownian map\, including those between exceptional points. This is based on
  joint work with Jason Miller (https://arxiv.org/abs/2008.02242).\n\nFirst
 \, 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.\n\nThen\, we show that the intersection of any two geodes
 ics minus their endpoints is connected\, the number of geodesics which ema
 nate from a single point and are disjoint except at their starting point i
 s at most 5\, and the maximal number of geodesics which connect any pair o
 f 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\, suc
 h pairs have dimension zero and are countably infinite. Further\, we class
 ify the (finite number of) possible configurations of geodesics between an
 y pair of points\, up to homeomorphism\, and give a dimension upper bound 
 for the set of endpoints in each case. \n\nFinally\, we show that every ge
 odesic can be approximated arbitrarily well and in a strong sense by a geo
 desic connecting typical points.  In particular\, this gives an affirmativ
 e answer to a conjecture of Angel\, Kolesnik\, and Miermont that the geode
 sic frame\, the union of all of the geodesics in the Brownian map minus th
 eir endpoints\, has dimension one\, the dimension of a single geodesic.  \
 n
LOCATION:Zoom
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