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Isaac Newton Institute for Mathematical Sciences

Random Geometry

12 Jan-3 Jul 2015

Organisers: Itai Benjamini (Weizmann Institute of Science), Nathanaël Berestycki (Cambridge), Jean-François Le Gall (Paris-Sud) and Scott Sheffield (Massachusetts Institute of Technology)

Scientific Advisory Committee: Geoffrey Grimmett (Cambridge), James Norris (Cambridge), Yuval Peres (Microsoft Research), Stanislav Smirnov (Geneva), Wendelin Werner (Paris, Orsay)

Programme Theme

RGM programme identifier: Flow lines of the Gaussian free field with different initial angles indicated by different colors emanating from the bottom of the squareA new frontier has emerged at the interface between probability, geometry, and analysis, with a central target to produce a coherent theory of the geometry of random structures. The principal question is the following: within a given structure, what is the interplay between randomness and geometry? More precisely, does the geometry appear to be random at every scale (i.e. fractal), or do fluctuations "average out" at sufficiently large scales? Can the global geometry be described by taking a suitable scaling limit that allows for concrete computations?

Spectacular progress has been made over the last ten years in this domain. The goal of the programme is to gather experts from probability, geometry, analysis and other connected areas, in order to study aspects of this question in some paradigmatic situations. Topics of particular relevance include the Gaussian Free Field, random planar maps and Liouville quantum gravity, in connection with conformally invariant scaling limits; spin glass models and branching random walks; percolation and random graphs; and random walks on graphs and groups in the case where the geometry is determined by some algebraic ambient structure.