This article describes new results concerning the geometry of Brownianmotion. Brownian motion is a model that describes the hectic, random jostlingof particles. The likelihood that the particles' paths will crossis measured by "intersection exponents", which are of special interestto physicists because of their importance in understanding phase transitions. New work concerning the intersection exponents has led to the discoveryof a new kind of random process, called stochastic Loewner evolution, whichmay prove extremely useful in physics. The work also settled a 1982conjecture of fractal pioneer Benoit Mandelbrot. He suggested thatthe length of the "frontier", or outer edge, of a Brownian path is proportionalto the diameter of the frontier (that is, the longest distance across thefrontier). Just as Mandelbrot predicted, the ratio has now been shownto be 4/3.
--- Allyn Jackson