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Where probability, geometry and analysis collide

We are all familiar with the Platonically perfect objects of ordinary geometry: circles, lines, spheres, planes, and so forth.  But are there Platonically perfect ways to describe random geometric structures? In other words, is there a Platonically perfect probability measure on the set of loops, or the set of curves, or the set of surfaces, etc.?  And if so, can one show that these perfect probability measures are limits of simple discrete probability measures (the way the perfect “bell shaped curve” is the limit of the probability measure describing the number of heads in n tosses of a fair coin)? How do these perfect probability measures somehow appear in the mathematical models inspired by string theory, statistical physics, finance, biology, and so forth? What interesting results and predictions does one obtain from this theory? These were some of the questions posed by the organisers of the highly successful 2015 programme on Random Geometry (RGM). 

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons