In the last dozen or so years several subject areas in physics and
mathematics have received a tremendous boost by focusing on a diverse group
of related problems of interest to both fields, and by sharing their
respective techniques and methods. These problems include critical phenomena
near second order phase transitions, random fields and quantum gravity,
random matrices, unstable fluid flows, diffusion limited aggregation, and
other growth models. Stochastic structures play crucial role in all these
problems. Thus they benefit tremendously from the use of probabilistic
methods. Another important common feature of these problems is an intrinsic
interplay of mathematical and physical approaches, methods, and intuition. A
prime example of this relation is the study of stochastic geometry of
critical clusters and interfaces in two dimensions, where methods of
conformal field theory, conformal maps, and probability theory were blended
into the beautiful theory of Schramm-Loewner Evolutions and related subjects.
I will present a (personally and professionally biased) overview of these
developments, and will outline the outstanding problems that will hopefully
be attacked in the near future by the same concerted effort of physicists and
mathematicians.
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