# Oded Schramm

### Born: 10 December 1961 in Jerusalem, Israel

Died: 1 September 2008 in Guye Peak, Washington State, USA

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**Oded Schramm**attended the Hebrew University of Jerusalem, the foremost university in Israel, where he obtained his bachelor's degree in mathematics and computer science in 1986. He continued to work at the Hebrew University under Gil Kalai and submitted his Master's thesis

*Borsuk's Problem and the Set of Middle Points of Diameters*, on 21 May 1987:-

After the award of a Master's degree, Schramm went to the United States to undertake work for his doctorate at Princeton University. He undertook research with William Thurston as his thesis advisor, and submitted his thesis... it is carefully hand written45pages(in Hebrew)with many drawings and footnotes.

*Packing Two-Dimensional Bodies With Prescribed Combinatorics And Applications To The Construction of Conformal and Quasi-Conformal Mappings*in 1990.

Before the award of his doctorate, Schramm had papers in print. For example, in

*On the volume of sets having constant width*(1988) he gave improved lower bounds on the minimum volume of an

*n*-dimensional set of constant width 1. In

*Illuminating sets of constant width*(1988) he looked at the minimum number of directions required to illuminate the entire boundary of an

*n*-dimensional body by sets of light rays parallel to these directions. The paper

*Existence and uniqueness of packings with specified combinatorics*(1991), containing work done for his doctoral thesis, gave a highly significant generalization of the Andreev-Thurston circle packing theorem.

Following the award of his doctorate, Schramm worked for two years as a postdoctoral fellow at the University of California, San Diego. In 1992 he returned to Israel when appointed to the Weizmann Institute in Rehovot, Israel. This Institute was founded by Chaim Weizmann, first president of Israel, in 1934. It was renamed the Weizmann Institute of Science in 1944. Schramm worked at the Institute for seven years until, in 1999, he returned to the United States to take up a position in the Theory Group at Microsoft Research in Redmond, Washington.

In 1994 Schramm published a joint paper with Greg Kuperberg entitled

*Average kissing numbers for non-congruent sphere packings*. Kuperberg gives this summary of the work:-

In 2000 Schramm publishedOded was looking at what graphs can be realized in^{n}by kissing spheres.(The spheres must have disjoint interiors, but they don't have to be the same size.)It's a standard theorem in classical geometry, also related to important work in hyperbolic geometry and complex analysis, that you can realize any planar simple graph by kissing circles in^{2}, i.e., the circles are the vertices and the kissing pairs are the edges. It's not clear what restrictions there are in^{n}forn> 2, but Oded observed that there clearly are some because you can show that the average kissing number for any finite configuration is at most twice the maximum kissing number for equal-sized spheres. Our paper shows that the average kissing number for a finite set of kissing spheres in^{3}is bounded above by a constant which is strictly less than15, but on the other hand can be more than12.

*Scaling limits of loop-erased random walks and uniform spanning trees*which introduced the important concept now named Schramm-Loewner evolution. The Clay Mathematics Institute gives this summary [3]:-

The list of awards and prizes he received shows the remarkable quality of his work. Among the first prizes he received were the Anna and Lajos Erdős Prize in Mathematics in 1996, and the Salem Prize in 2001 (jointly with Stanislav Smirnov):-While Schramm is known in many fields, he is perhaps most widely recognized for his ground-breaking proposal of stochastic Loewner evolution, or SLE. This work led to the solution of many problems by him, many together with his collaborators Greg Lawler now at Cornell University, and Wendelin Werner in Strasbourg, France, as well as by many other mathematical researchers. His work in a spectacular series of papers has led to major progress in probability theory, in the theory of percolation and of random walks, as well as in related topics of conformal field theory.

In the following year he received the 2002 Clay Research Award for:-Schramm was recognized for his development of stochastic Loewner equations and for his contributions to the geometry of Brownian curves in the plane.

He then received the Loève Prize in 2003. David Aldous writes [1]:-... work in combining analytic power with geometric insight in the field of random walks, percolation, and probability theory in general, especially for formulating stochastic Loewner evolution. His work opens new doors and reinvigorates research in these fields.

Next he received the Henri Poincaré Prize in 2003:-His research in probability was sparked by his interest in the conjecture that the limit of two-dimensional critical percolation was conformally invariant. In trying to understand this limit as well as limits of other models such as the loop-erased walk, Schramm combined classical results in complex variables of C Loewner with probability theory to invent the process now called the Schramm-Loewner evolution(SLE). This process has proved to be a critical ingredient for understanding conformally invariant limits of planar systems. In collaboration with G Lawler and W Werner, Schramm has used SLE to solve a number of open problems, in particular Mandelbrot's conjecture that the outer boundary of planar Brownian motion has dimension4^{}/3_{}and the determination of the scaling limit of loop-erased work. Schramm also showed that if the scaling limit of percolation was conformally invariant, then the boundaries between clusters would be given by SLE. That this is true for site percolation on the triangular lattice has been proved by S Smirnov.

He was invited to give a plenary addressFor his contributions to discrete conformal geometry, where he discovered new classes of circle patterns described by integrable systems and proved the ultimate results on convergence to the corresponding conformal mappings, and for the discovery of the Stochastic Loewner Process as a candidate for scaling limits in two dimensional statistical mechanics.

*Conformally invariant random processes*at the 2004 European Congress of Mathematics in Stockholm, Sweden. Here is the abstract he gave for his lecture:-

He was also invited to give the prestigious Coxeter Lecture Series at the Fields Institute in September 2005. He gave the following abstract for his three lecture series onWe will survey and explain a recent theory describing precisely the scaling limits of many random systems in two dimensions. Random paths associated with each of these systems are believed to converge to a path among a one-parameter family of random fractal curves called stochastic Loewner evolution(or SLE). Several instances of this statement have been proven, for example, critical percolation on the triangular grid and loop-erased random walks, while others are still conjectural, e.g., the Ising and Potts models and the self-avoiding walk. The theory is useful, mainly because the SLE description facilitates explicit calculations of properties of the scaling limits.

*Scaling limits of two dimensional random systems*:-

He was awarded the SIAM George Pólya Prize in 2006 which he received jointly with Gregory Lawler and Wendelin Werner:-The simple random walk on the square grid in the plane converges to Brownian motion under appropriate scaling. Planar Brownian motion is rotationally invariant, while the random walk is not. In fact, Brownian motion enjoys conformal invariance, which is far richer. Many other random systems, such as critical percolation and the critical Ising model of magnetism, also seem to exhibit conformal invariance in the scaling limit. This conformal invariance implies that certain paths arising naturally from these processes fall into a particular one-parameter family of random fractal paths called Stochastic Loewner Evolution, or SLE.

He was invited to give a plenary address the 2006 International Congress of Mathematicians in Madrid. His address... for their groundbreaking work on the development and application of stochastic Loewner evolution(SLE). Of particular note is the rigorous establishment of the existence and conformal invariance of critical scaling limits of a number of2D lattice models arising in statistical physics.

*Random, Conformally Invariant Scaling Limits in 2 Dimensions*was advertised as follows:-

Prestigious awards continued to he presented to Scramm, for he received next the Ostrowski Prize in 2007. In 2008 he was elected as a foreign member of the Royal Swedish Academy of Sciences.Nuclear reactions, living organisms, sociological studies... For many years physicists have used predictions to describe the behaviour of these systems. Only recently have mathematical proofs been found for these predictions. In his plenary lecture at the ICM2006, Oded Schramm will discuss various open problems, some of which have been selected to illustrate achievements in physics that have yet to be understood mathematically. Schramm will speak about random systems in the plane, and about how to understand their behaviour. One example of a random system is the so-called percolation. Percolation analyses the movement of a fluid in a porous material. For oil and natural gas, for instance, theoretical results in this area have helped to improve productivity in the exploitation of both these resources. Although some of the systems studied by Professor Schramm are well defined mathematically, he points out that "often arguments provided by physics cannot be translated into mathematical arguments that make sense."

Kenneth Chang explains in [2] that Schramm almost certainly only missed the award of a Fields Medal due to a technicality:-

Schramm, who was an avid climber, died in a mountaineering accident at Guye Peak, near Snoqualmie Pass, in Washington State. He had gone climbing on his own and when he failed to return as planned on 1 September 2008, his wife reported him missing. His body was discovered the next morning. He leaves a wife Avivit and two children, Tselil and Pele. In addition to the long list that we detailed above of prestigious honours he had received, he was due to give the Gibbs Lecture in 2009.If Dr Schramm had been born three weeks and a day later, he would almost certainly have been one of the winners of the Fields Medal, perhaps the highest honour in mathematics, in2002. But the Fields Medals, which honour groundbreaking work by young mathematicians, are awarded only once every four years and only to mathematicians who are40or under. Dr Schramm was born on December10,1961; the cut-off birth date for the2002Fields was January1,1962. Wendelin Werner, a younger mathematician who collaborated with Dr Schramm on follow-up research, won a Fields in2006.

**Article by:** *J J O'Connor* and *E F Robertson*

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