Institute for Applied Mathematics  / Probability Theory and Stochastic Analysis  / Teaching
University of Bonn Institute for Applied Mathematics
 
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    S4F2 Graduate Seminar in Stochastic Analysis:

    Random maps and su    faces

     

                (picture by J. Bettinelli)

     


    Time: Wednesdays 14-16

    Room:  SR 1.008

     

    Description:

    "What is a natural random geometry on a given surface"

    That is the question that we want to discuss in the seminar. Over the past few decades, two major classes of random surface models have emerged within physics and mathematics. One is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics. Only very recently there are exiting developments linking the two approaches.

    We will focus on the second approach, which is based on discretization of the surface choosing uniformly at random among all quadrangulations of the surface with $n$ faces, called planar maps. Just as Brownian motion is the universal scaling limit of random walks in 1D, we will see that these random planar maps admit a universal scaling limit, the so-called Brownian map, which is a random metric space with fascinating fractal properties. Along the  way, we will develop beautiful combinatorial tools to study planar maps, in particular they can be encoded by certain labeled trees. Brownian motion/excursions and the continuum limit of these random trees will play a major role in identifying the universal scaling limit, i.e.~the random geometry on the surface that we are looking for.

     

    Prerequisites:

    solid background in probability and stochastic processes

     

    Schedule:

    Date Topic Speaker
    30.10. Background on planar maps Matthias
    06.11. Combinatorics of planar maps and the CVS bijection Agnes
    13.11. Scaling limits of random trees I Hanjo
    20.11. Scaling limits of random trees II Hanjo
    27.11. First scaling limit results for planar maps Koen
    04.12. The Brownian map I Aleksandra
    11.12. The Brownian map II Aleksandra
    18.12. Geodesics in planar maps I Adrian
    08.01. Geodesics in planar maps II Adrian
    15.01. Uniqueness of the scaling limit Fabian
    22.01. Universality of the Brownian map NN

     

    Reference:

    • G. Miermont, Aspects of  Random Maps, Lecture notes of the 2014 Saint-Flour Probability Summer School [pdf]