# Mathematics Alumni

Werner was born in Germany in 1968, but is of French nationality. He is currently at the University of Paris-Sud in Orsay.

His research is motivated by problems in statistical mechanics, the branch of theoretical physics that deals with bulk properties of matter—solids, liquids, gases—in terms of atomic structure. One of his most dramatic discoveries, confirming a conjecture of Benoit Mandelbrot, the founder of modern fractal geometry, relates to the earliest observational evidence supporting the theory that everything is made from atoms.

This is the phenomenon of Brownian motion. In 1827 the botanist Robert Brown in 1827 was looking through his microscope at pollen particles floating in water. He noticed that tiny particles inside the grains were jiggling around at random. The same effect arose using dust particles, so it had nothing to do with pollen being part of a living organism. Later, this motion was explained by Louis Bachelier, Albert Einstein, and others as the result of the particles repeatedly being hit by much smaller, invisible, atoms.

A mathematical model of Brownian motion was developed by Norbert Wiener in the 1930s. One way to describe it (not the actual technical definition) is to make the particle perform a ‘random walk’, repeatedly moving a tiny distance along some random direction to form a wiggly path. As the step size tends to zero, and with the right statistical assumptions, the result produces Wiener’s model.

Brownian motion

Brownian paths are continuous curves, but they are ‘almost always’ non-differentiable at every point—that is, it has no well-defined tangent.

The form of the curve is fractal, so the basic question of its fractal dimension arises. Mandelbrot made a conjecture about the fractal dimension of a typical Brownian path—the wiggly curve traced by the randomly moving particle. This curve may cross itself at some points. If so, fill in the resulting hole. The boundary of this filled-in shape is the outer boundary of the Brownian path. Mandelbrot conjectured that the fractal dimension of the outer boundary is exactly 4/3.

In 2000, Werner, working with Gregory Lawler and Oded Schramm, proved Mandelbrot’s conjecture, and established several other fractal properties of Brownian motion as well. Their proof involves mathematical analogies with other fractal processes, notably the theory of ‘percolation’. Other mathematical physicists have suggested different proofs based on links with quantum gravity.

Other aspects of Werner’s work include quantitative features (‘critical exponents’) of phase transitions, such as the change from liquid to gas, and detailed features of the geometry of various statistical-mechanical processes (‘conformal invariance’).