Oded Schramm Memorial Conference

Probability and Geometry

August 30-31, 2009

Held at Microsoft Research.


This was a two-day conference held in honor of Oded Schramm and his mathematics, and took place in Building 99 at Microsoft Research in Redmond, Washington. The talks were recorded and are available here, together with slides, by clicking on the images in the schedule. The recordings are by session rather than by talk, but it is possible to fast-forward. The recordings are viewable by users of Internet Explorer and Windows Media Player; we will make the slides available independent of the lectures for the benefit of other users (please check back later).

There is an upcoming Midrasha Mathematicae - Discrete Probability and Geometry: The mathematics of Oded Schramm. It will be in Jerusalem, December 15-23 2009. For further information contact Itai Benjamini.

Schedule

Sunday August 30
9:00 - 9:30 refreshments
9:30 - 9:35 Jennifer Chayes (Microsoft)
Opening
9:40 - 10:30 Wendelin Werner (Orsay)
Oded and his SLE processes
coffee break
11:00 - 11:30   Ken Stephenson (U. Tennessee)
Rigidity of circle packings [slides]
11:35 - 12:05 Zheng-Xu He (Beijing CAS)
Disk packings and conformal maps
12:10 - 12:40 Igor Pak (UMN and UCLA)
Caged eggs and the rigidity of convex polyhedra
lunch (catered)
2:00 - 2:30 Olle Häggström (Chalmers)
Percolation, mass transport and cluster indistinguishability
2:35 - 3:05 Yuval Peres (Microsoft)
Connectivity Probability in Critical Percolation:
An unpublished gem from Oded
coffee break
3:35 - 4:05 David B. Wilson (Microsoft)
Oded's work on Boolean functions
4:10 - 4:40 Christophe Garban (ENS Paris)
Oded's work on Noise Sensitivity [slides]
4:45 - 5:15 Gábor Pete (U. Toronto)
How to prove tightness for the size of strange random sets
6:00 - 9:00 Conference dinner at Matt's Rotisserie and Oyster Lounge [directions].


Monday August 31
9:00 - 9:30 refreshments
9:30 - 9:35 Rico Malvar (Microsoft)
A few words from the director of MSR-Redmond
9:35 - 10:05 Omer Angel (U. British Columbia)
Random planar maps and their limits
10:10 - 10:40 Michael Freedman (Microsoft)
Random triangulations as dynamical variables in quantum mechanical models
coffee break
11:05 - 11:35 Russell Lyons (Indiana U.)
Hyperfinite graph limits
11:40 - 12:10 Steffen Rohde (U. Washington)
Jordan Curves and Dimension of Quasicircles
lunch (cafeteria)
1:15 - 1:30 Reminiscences
1:30 - 2:00 Mario Bonk (U. Michigan)
Transboundary Extremal Length
2:05 - 2:35 Gregory F. Lawler (U. Chicago)
Path Properties of Schramm-Loewner Evolution (SLE) [slides]
coffee break
3:00 - 3:50 Stanislav Smirnov (U. Genčve)
SLE, percolation, and scaling limits
3:55 - 4:25 Scott Sheffield (MIT)
SLE as a scaling limit and the Gaussian free field
4:30 Conference concludes

Abstracts

Random planar maps and their limits
Omer Angel
Abstract: I will survey the area of random planar graphs and their limits, including the Benjamini-Schramm result on recurrence of the limits, its extension to excluded minor graphs, as well as existence and some basic properties of the limit of uniform planar maps.

Transboundary Extremal Length
Mario Bonk
Abstract: In 1995 Oded introduced the concept of Transboundary Extremal Length. It is a generalization of classical extremal length in conformal geometry and very useful in the study of conformal mappings of multiply connected domains. In my talk I will review Oded's work on the subject and indicate some future directions.

Random triangulations as dynamical variables in quantum mechanical models
Michael Freedman
Abstract: I'll discuss a simple calculation which Oded showed me how to do and its implications for taking quantum mechanical models off lattice.

Oded's work on Noise Sensitivity [slides]
Christophe Garban
Abstract: In this talk, I will survey different techniques developed by Oded (et al) which enable to study the Spectral decomposition of Boolean functions. The focus will be mostly on percolation. We will see that in some sense, critical percolation is a system whose large scale properties are of "High frequency". Among the different approaches investigated by Oded, I will describe one in more details: his work with Jeff Steif which studies the Fourier Spectrum of Boolean functions with the help of randomized algorithms.

Percolation, mass transport and cluster indistinguishability
Olle Häggström
Abstract: Oded Schramm had huge impact on the study of percolation beyond Zd. I will discuss two of the main tools - mass transport and stationarity of delayed random walk - employed by him and his coauthors, and how these were used by Lyons and Schramm to obtain a remarkable result known as cluster indistinguishability.

Disk packings and conformal maps
Zheng-Xu He
Abstract: We will review some joint works on the convergence of disk packing solutions to the conformal maps; in particular the convergence of second order derivatives of disk packing solutions with unrestricted combinatorial pattern. The convergence of higher order derivatives will be briefly discussed.

Path Properties of Schramm-Loewner Evolution (SLE) [slides]
Gregory F. Lawler
Abstract: The Schramm-Loewner evolution, invented by Oded Schramm, is a fascinating mathematical object. I will discuss some of the arguments used to understand the curve rigorously with an emphasis on path properties.

Hyperfinite graph limits
Russell Lyons
Abstract: Elek called a collection of finite graphs hyperfinite if its graphs can be decomposed into pieces of uniformly bounded size by deletion of a uniformly and arbitrarily small density of edges. A unimodular probability measure on rooted graphs is called amenable if it can be decomposed in a unimodular way into finite pieces by deleting an arbitrarily small density of edges. It is easy to see that every limit point of a hyperfinite collection is amenable. We sketch Oded's proof of the converse that every bounded-degree sequence with amenable limit is necessarily hyperfinite. He applied a quantitative of this with Itai Benjamini and Asaf Shapira to property testing in computer science.

Caged eggs and the rigidity of convex polyhedra
Igor Pak
Abstract: In his celebrated "How to cage an egg" paper, Oded Schramm found a remarkable result generalizing in an unusual direction the circle packing theorem, the Steinitz theorem and a number of other related results. In this talk I will discuss the proof which is somewhat technical but actually quite beautiful, and is based on the ideas behind the rigidity results by Cauchy and Alexandrov. I will present the general outline of the proof, and then show how Dehn's rigidity theorem can be obtained along similar lines using Oded Schramm's tools.

Connectivity Probability in Critical Percolation: An unpublished gem from Oded.
Yuval Peres
Abstract: Let G be a nonamenable Cayley graph (for all finite sets in G, the surface to volume ratio is bounded away from 0) and let X0, X1, X2,... be simple random walk on a G. I will relate Oded's proof that for critical percolation on G, the probability that there is an open path between X0 and Xk decays exponentially in k. While the proof is short, and all the ingredients were known to the percolation community, the recipe that Oded found to combine them is dazzling, and I remember (almost) falling off my chair when Oded first showed me his argument. No background will be assumed on amenability or critical percolation.

How to prove tightness for the size of strange random sets
Gábor Pete
Abstract: If we want to show that a random set is unlikely to be much smaller than expected, but only very limited independence information is available, what shall we do? I will discuss a strategy that Oded designed in his work with Christophe Garban and myself to deal with the Fourier spectral sample of critical percolation, involving a new large deviation inequality for dependent variables.

Jordan Curves and Dimension of Quasicircles
Steffen Rohde
Abstract: This talk is about some old (mid 90's) yet-to-be published work of Oded, Kari Astala, and myself concerning conformal geometry. Quasiconformal maps are important generalizations of conformal maps that appear naturally in many settings. In his celebrated paper on Area distortion of quasiconformal mappings (Acta Math. 173, 1994), Kari conjectured that the maximal Hausdorff dimension d(k) among all k-quasicircles is 1+k2, for all k between 0 and 1. The upper bound of the conjecture, d(k)≤1+k2, was proved by Smirnov (Acta Math., to appear). We studied self-similar simple curves, found a lower bound for d(k), and showed that a certain linear analog of the Mandelbrot set is disconnected.

SLE as a scaling limit and the Gaussian free field
Scott Sheffield
Abstract: I will discuss Oded's perspective on Loewner driving functions, martingale observables, and Gaussian free field couplings.

SLE, percolation, and scaling limits
Stanislav Smirnov
Abstract: I will discuss how Oded's work on SLE and beyound improved our uderstanding of the percolation scaling limit.

Rigidity of circle packings [slides]
Ken Stephenson
Abstract: A "circle packing" is a configuration of circles satisfying some specified pattern of tangencies. A first surprise is the wealth of examples --- there's a circle packing associated with any triangulation of any topological surface. A deeper surprise lies in their uniqueness, for the rigidity of circle packings is the rigidity of analytic functions in "quantum" form. We will recognize Oded's unique mathematical style in some (quite unbelievable!) existence results, in a new proof of the Riemann Mapping Theorem, and in a wonderfully elementary proof of uniqueness. I will sketch the latter, which for me is the most beautiful proof in mathematics.

Oded and his SLE processes
Wendelin Werner
Abstract: I will try to remember some episodes of the time when Oded's SLE processes entered in our life.

Oded's work on Boolean functions
David B. Wilson
Abstract: I will introduce some of Oded's work on Boolean functions and decision trees, and discuss how it relates to random-turn Hex and percolation.

Participants

David Aldous
Reid Andersen
Omer Angel
Tonci Antunovic
Robert Bauer
Tomek Bartoszynski
Itai Benjamini
Nathanael Berestycki
Mariusz Bieniek
Ilia Binder
Nathaniel Blair-Stahn
Horatio Boedihardjo
Mario Bonk
Christian Borgs
Krzysztof Burdzy
Jennifer Chayes
Zhen-Qing Chen
Dayue Chen
Amir Dembo
Jian Ding
Mauricio Duarte
Julien Dubédat
Ioana Dumitriu
Bertrand Duplantier
Vance Faber
Wai Tong Fan
Abie Flaxman
Steven Flores
Michael Freedman
Christophe Garban
Subhro Ghosh
James Gill
Jesse Goodman
Vadim Gorin
Ilya Gruzberg
Ori Gurel-Gurevich
Olle Häggström
Alan M. Hammond
Zheng-Xu He
Chris Hoffman
Ander Holroyd
Kamal Jain
Yicheng Kang
Anna Kazeykina
Nathan Keller
Richard Kenyon
Julia Komjathy
Michael Kozdron
Gregory F. Lawler
Lionel Levine
Joan Lind
Eyal Lubetzky
Russell Lyons
Peter March
Don Marshall
Robert Masson
Sergei Merenkov
Daniel Meyer
Jason Miller
Andrey Mishchenko
Ben Morris
Elchanan Mossel
Asaf Nachmias
Seffi Naor
Pierre Nolin
Igor Pak
Yuval Peres
Gábor Pete
James Propp
Steffen Rohde
Dan Romik
Julia Ruscher
Tom Salisbury
Pele Schramm
Scott Sheffield
Vladas Sidoravicius
Allan Sly
Stanislav Smirnov
Boris Solomyak
Terry Soo
Ken Stephenson
Nike Sun
Vincent Tassion
Prasad Tetali
Joshua Tokle
Huy V. Tran
Jonathan Tsai
Brigitta Vermesi
Bálint Virág
Wendelin Werner
David White
David B. Wilson
Carto Wong
Dapeng Zhan

Directions

From the north: Take I-5 south, then I-405 south, then WA-520 east.
From the south: Take I-5 north, then I-405 north, then WA-520 east.
From Seattle: Take WA-520 east.
By airplane: Fly to Seattle's airport, take I-405 north, then WA-520 east.

From WA-520 east, take the 148th Ave NE North exit (this is the second 148th Ave NE exit). Turn right (north) onto 148th Ave NE, proceed a few blocks, and turn right onto NE 36th St. Building 99 will be on the left. The address is 14820 NE 36th St, Redmond, WA 98052-5319. Click here for a map.

From the Homestead Studio Suites in Bellevue, one can either walk to Microsoft Research (about 1/2 hour) or drive. To walk, the recommended route is to turn right (north) on 156th Ave NE, left on 40th St, left on 150th Ave NE, and right on NE 36th St. This route also works driving, but a slightly shorter route would be to turn left (south) on 156th Ave NE, right on BelRed Rd, right on NE 24th St, right on 148th Ave NE, and right on NE 36th St. (Refer to the map.) On the second day (Monday), it is also possible to walk across the street to Microsoft and ask for a shuttle to building 99.

Hotels

Nearby hotels include

Poster

Poster for the conference

Related

Wikipedia page on Oded Schramm
Oded Schramm Memorial web site (includes some of Oded's talks)
Oded Schramm Memorial blog
Terry Tao's blog on Oded Schramm
Gil Kalai's blog on Oded Schramm
Oded Schramm's home page
MathSciNet information on Oded's articles

Midrasha Mathematicae - Discrete Probability and Geometry
The mathematics of Oded Schramm

Jerusalem
15-23 December 2009
The school will be devoted to discrete probability and its relations with geometry and other areas of mathematics. The topics are chosen to represent the research interests of Oded Schramm. For further information contact Itai Benjamini.




File:CircleRiemannMap1.svg