Ity and Geometry the Mathematics of Oded Schramm

Midrasha Mathematicae - Discrete Probabil- ity and Geometry The mathematics of Oded Schramm

The Midrasha (school) took place from 15 to 23 December 2009, at the Hebrew University of Jerusalem. The lectures where given at the Feldman building, Givat Ram and the school was hosted and supported by the Institite for Advanced Studies of the Hebrew University of Jerusalem (HUJI-IAS).It further supported by The Clay Mathematical Institute, the Weizmann In- stitute, the Mathematics Institute of the Hebrew University of Jerusalem, Microsoft Research, and individual grants.

Topic The school was devoted to probability and its relations with geometry and other areas of mathematics. The topics were chosen to represent the research interests of Oded Schramm.

Academic directorship Peter Sarnak (Princeton University), general director for the Midrasha pro- gram; Gil Kalai, Noam Berger (The Hebrew University),Itai Benjamini (The Weizmann Institute), and Werner Wendlin (Universit Paris-Sud), directors of the 2009 Midrasha; Eliezer Rabinovici, director of HUJI-IAS; David A. Elwood, academic director of the Clay Mathematical Institute.

Financial support • HUJI-IAS through the Rothschild memorial foundation and other sources 30,000 (approximated).

• Clay Mathematics Institute 10,000

• Weizmann Institute 5,000

• Microsoft Research 5,000

1 • Institute of Mathematics at HU 5,000

• Individuals Research grants 10,000 (approximated)

Lecturers and participants There were 27 lecturers (6 from Israel, 11 from Europe, 9 from the U.S. & Canada, 1 from South America). There were 59 registered participants (31 from abroad: Canada and the US 11, Asia 1, Europe 19, 28 from Israel) and around 20-30 other participants (most from Israel).

Two research highlights Although this was a school and not a research conference some talks reported on very recent research highlights. I will mention two: Scott Sheﬃeld reported on his work with Jerison and Levine that settled a central prob- lems on Internal DLA, Stanislav Smirnov reported on a recent breakthough regarding self-avoiding random walk in the plane.

Lectures divided into areas

There were altogether 37 lectures.

Basic stochastic geometric and combinatorial models Several lectures introduced and discussed basic stochastic geometric models. Yuval Peres gave a talk about percolation, random turn Hex, and Boolean function; His second lecture was about sand pile models and DLA. Ori Gurel- Gurevich presentad a new but very fundamental result about random walks, David Wilson discussed ﬁnitary colorings and Vladas Sidoravicius talked about models of Markov chains. Scott Sheﬃeld second lecture dealt with the solution of the Internal DLA model. Elchanan Mossel discussed stochastic models of elections and voter manipulation. Miklos Albert and Martin Zerner described important aspects of percolation. Nathanel Berestycki discussed mixing times for carious statistical physics models.

2 SLE SLE - stochastic Lowener equations (or “Schramm Lowener equations) are re- markable classes of planar stochastic models introduced by Oded Schramm. Greg Lawler gave a colloquium lecture introducing these models, Beffara gave a series of two lectures on their dimension, Stanislav Smirnov gave a lecture on their relation to discrete models, and Scott Sheffield’s first lecture described a new techniques for relating discrete planar models to SLE. Wen- delin Werner’s lecture on Bessel processes was also related to SLE. Eldad Battelheim, a HU physicist explained how SLE occurs in models of quantum statistical physics related to condensed matter physics and gave some insight in which cases SLE computation can answer questions not treated by more traditional computational methods of conformal field theory. Greg Lawler gave some additional talks on SLE for a few interested students.

Limits of graphs Laszlo Lovasz gave a series of two lectures on limit of graphs, Balasz Szegedy gave two lectures on the connection between graph limits and extremal com- binatorics, and Gabor Elek gave another series of two lectures explaining other aspects of this theory. Omer Angel described connection between scaling limit of random planar triangulation and circle packings. Nicolas Curien explain how to study the scaling limit of planar maps and create the “Brownian cactus” based on basic enumeration theory and related probability.

Noise sensitivity, dynamic percolation,and analysis of Boolean function Jeffery Steiff and Cristophe Garban gave a series of four lectures on noise sensitivity, spectrum and Dynamical percolation, Gabor Pete gave an additional lecture describing his recent definite work ith Schramm and Garban in this direction. Stanislav Smirnov showed his proof with Schramm that percolation is a black noise. Boris Tsirelson’s talks used quantum probability method to show that certain evolutions in physics (more precisely, stochastic evolutions which admit certain unitary representations) are necessarily stochastically-classic (noise-stable). Nathan Keller described how to prove

3 a sharp form of a theorem by Benjamini, Kalai, and Schramm about noise sensitivity and inﬂuences.

Percolation on nonamenable groups . Gady Kozma gave a lecture on percolation on non amenable groups, an area introduced by Benjamini and Schramm and explained new “non- lace/non-perturbative” methods to prove the triangular condition needed to show a mean free ﬁeld behavior.

Concluding lecture Joram Lindenstrauss described, in his closing lecture, some works by Oded Schramm on covering convex sets with sets of smaller diameter, and on sets of constant width. He described a subsequent work with Bourgain about covering sets with spheres.