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Conformal Geometry

Conformal Geometry

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Organized by Ilia Binder, John Cardy, Andrei Okounkov, and Paul Wiegmann
January 7 – May 3, 2013

The Simons Center will host a program on `Conformal Geometry’ for the Spring
semester of 2013. This will cover subjects representing some of the most successful
examples of the cross-fertilization between mathematics and physics in this
century. It has led to the award of three Fields Medals in mathematics and a
resurgence of interest in conformal field theory (CFT) and other areas of physics
and mathematics where conformal symmetry emerges. Prior to these major
developments conformal symmetry had been in explicitly observed only in a few
`one-dimensional’ random processes: Brownian motion, random trees, and the like.
Inspired by work of physicists in the 70s and 80s on conformal invariance and field
theories in two dimensions, a number of leading probabilists and combinatorialists
began thinking about spatial process in two dimensions: percolation, polymers,
dimer models, Ising models. Major breakthroughs by Kenyon, Schramm, Lawler,
Werner, Smirnov, Sheffield and others led to a rigorous underpinning of conformal
invariance in two-dimensional systems and paved the way for a new era of two-
dimensional probability theory. In recent years probability theory (and here we
mean probability theory in the largest sense, comprising combinatorics, statistical
mechanics, algorithms, simulation) has made immense progress in understanding
the basic two-dimensional models of statistical mechanics and random surfaces. At
the same time deep links have emerged with various manifestations of integrable

Conformal Field Theory, as originally developed to describe critical phenomena,
focuses on the operator algebra of local operators and representations of infinite-
dimensional algebras. Conversely the probability theory of conformal processes
focuses on stochastic processes and random objects which have a geometric
interpretation. Together, these problems constitute a newly emerging field of
conformal stochastic geometry, or simply `Conformal Geometry.’

This emergent field of Conformal Geometry establishes new interesting links
and synergy between probability theory, geometry, representation theory and
analysis. The methods emerging within this field have proved to be efficient in many
important applications in physics. These range across many different physics sub-
disciplines: from traditional models of statistical mechanics, disordered systems,
quantum gravity, and random matrices, to quantum and classical non-equilibrium
phenomena and fluid mechanics.