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
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.
Sunday August 30
Random planar maps and their limits
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
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
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]
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
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
Disk packings and conformal maps
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
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
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.
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
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
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
Abstract: I will discuss Oded's perspective on Loewner driving functions, martingale observables, and Gaussian free field couplings.
SLE, percolation, and scaling limits
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]
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
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
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.
David Aldous |
Wai Tong Fan
James Gill |
Alan M. Hammond
Gregory F. Lawler
Huy V. Tran
David B. Wilson
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.
Nearby hotels include
Poster for the conference
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
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.