I am very saddened (and stunned) to learn that Oded Schramm, who made fundamental contributions to conformal geometry, probability theory, and mathematical physics, died in a hiking accident this Monday, aged 46. (I knew him as a fellow editor of the Journal of the American Mathematical Society, as well as for his mathematical research, of course.) It is a loss of both a great mathematician and a great person.
One of Schramm’s most fundamental contributions to mathematics is the introduction of the stochastic Loewner equation (now sometimes called the Schramm-Loewner equation in his honour), together with his subsequent development of the theory of this equation with Greg Lawler and Wendelin Werner. (This work has been recognised by a number of awards, including the Fields Medal in 2006 to Wendelin.) This equation (which I state after the jump) describes, for each choice of a parameter , a random (fractal) curve in the plane; this random curve can be viewed as a nonlinear variant of Brownian motion, although the SLE curves tend to cross themselves much less frequently than Brownian paths do. By the nature of their construction, the curves are conformally invariant: any conformal transformation of an curve (respecting the boundary) gives another curve which has the same distribution as the original curve. (Brownian motion is also conformally invariant; given the close connections between Brownian motion and harmonic functions, it is not surprising that this fact is closely related to the fact that the property of a function being harmonic in the plane is preserved under conformal transformations.) Conversely, one can show that any conformally invariant random curve distribution which obeys some additional regularity and locality axioms must be of the form for some .
The amazing fact is that many other natural processes for generating random curves in the plane – e.g. loop-erased random walk, the boundary of Brownian motion (also known as the “Brownian frontier”), or the limit of percolation on the triangular lattice – are known or conjectured to be distributed according to for some specific (in the above three examples, is 2, 8/3, and 6 respectively). In particular, this implies that the highly non-trivial fact that such distributions are conformally invariant, a phenomenon that had been conjectured by physicists but which only obtained rigorous mathematical proof following the work of Schramm and his coauthors.
The Loewner equation was introduced by Charles Loewner in the 1920s as a means of generating conformal maps, which he then used to make progress on the Bieberbach conjecture. This equation can be phrased in any simply connected domain in the complex plane (which are all conformally equivalent anyway, thanks to the Riemann mapping theorem), but let us work with the upper half plane for simplicity. The starting observation is that for any , the map is a conformal map from the slit half-plane to . (Indeed, by the Schwarz lemma, this is the unique such conformal map which behaves like the identity near infinity.) Translating this, for any and , we have a conformal map from the slit half-plane to .
Now suppose that t=dt is infinitesimally small. Then by Taylor expansion, this conformal map is approximately given by the formula , thus each point z moves infinitesimally by the ODE
Infinitesimally, this ODE conformally deforms the half-plane by deleting an infinitesimal slit at w.
Now we work non-infinitesimally, letting t run from 0 to infinity, and letting vary in continuously in time. This (formally, at least), creates a time series of conformal maps from a half-plane with a curve removed to the full half-plane, which can be viewed as being formed by composing infinitely many of the preceding infinitesimal conformal transformations together. The flows are given by the Loewner equation
(The curve is basically the collection of initial data for which the above ODE degenerates by time t.)
Schramm observed that if the driving function was given by a multiple of one-dimensional Brownian motion for some , then the resulting (random fractal) curves continue to be well-defined (despite the non-differentiability of ) and become conformally invariant for any , although for the curve is no longer simple, and for in fact becomes space-filling). This opened the way to an enormous further literature on this subject by Lawler-Schramm-Werner and many others (see for instance this survey of Lawler).