Type: Seminar

Distribution: World

Expiry: 17 Jan 2014

CalTitle1: Limit shapes for random surfaces

Calendar2: 17 Jan 2014 1130-1230

CalLoc2: AGR Seminar: Werner

CalTitle2: AGR Seminar: Phase transitions and conformal invariance within planar fractal carpets

Auth: mathas@114.88.broadband5.iol.cz in SMS-auth

**Host venue**

AMSI

**Abstract**

I'll discuss the analytic solution to the limit shape problem for random domino tilings and "lozenge" tilings, and in particular try to explain how these limiting surfaces develop facets.

**Speaker biography** Professor of Mathematics Richard Kenyon interests are statistical mechanics, probability and discrete conformal geometry. He's also managed to think and talk about the higher elements of mathematics in two languages: English and French, with a "Habilitation thesis" from the Universite Paris-Sud (Sur la dynamique, la combinatoire, and la statistique des pavages) and a Ph.D. from Princeton in mathematics. His work has taken him from Orsay, France, where he was the research director at the Centre National de Recherche Scientifique, to Redmond, Wash., where for four summers he was a visiting researcher in the Microsoft theory group. Kenyon says he is driven "by the possibilitKenyon says tion and discovery. In the field of probability in particular, people have recently discovered many somewhat mysterious connecKenyon says he is driven "by the possibilitKenyon says tion and discovery. In the field of probability in particular, people have recently discovered many somewhat mysterious connecKeny so simple that it becomes mathematics." His prizes and awards have included the Clay foundation Senior Scholar, 2012, William R. Kenan Professorship, 2009, Prix Charles-Louis de Saulses de Freycinet from the French Academie des Sciences in 2002 and the Rollo Davidson prize in 2001. But what he's most proud of is "understanding the fluctuations of a certain model of random surfaces (called the dimer model). Basically it is a way to draw a ‘random' contour map or topographic map. You draw such a map and then ask questions such as how high, typically, is the highest mountain on this map. It turns out that this simple model has a certain property called ‘conformal invariance,' which was predicted by physicists in the 1970s but confirmed only recently." Kenyon is full professor in the mathematics department at Brown University since 2007.

It is now known for a number of models of statistical physics in two dimensions (such as percolation or the Ising model) that at their critical point, they do behave in a conformally invariant way in the large-scale limit, and do give rise in this limit to random fractals that can be mathematically described via Schramm's Stochastic Loewner Evolutions. The goal of the present talk will be to discuss some aspects of what remains valid or should remain valid about such models and their conformal invariance, when one looks at them within a fractal-type planar domain. We shall in particular describe (and characterize) a continuous percolation interface within certain particular random fractal carpets. Part of this talk will be based on joint work with Jason Miller and Scott Sheffield.

**Speaker Biography** Wendelin Werner (born September 23, 1968) is a German-born French mathematician working in the area of self-avoiding random walks, Schramm-LoeWendelin Werner (born September 2ies in probability theory and mathematical physics. In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain he received the Fields Medal. He is currently professor at ETH Zürich. He has received other awards, including the Fermat Prize in 2001, the Loève Prize in 2005, and the 2006 SIAM George Pólya Prize with his collaborators Gregory Lawler and Oded Schramm. He was awarded the Rollo Davidson Prize in 1998 and is a trustee of the Rollo Davidson Trust. He became a member of the French Academy of Sciences in 2008. He also had a part in the 1982 French film La Passante du Sans-Souci.

If you want to participate in either or both of these seminars contact Maaike Wienk at agr@amsi.org.au, with a cc to Michael Shaw at AMSI (mshaw@amsi.org.au), one week in advance at the latest.

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If you would like to attend this seminar in our access grid room then please book the access grid room referring to this scnews item. Please liaise with the host institution to make any necessary arrangements and then send an email to accessgridroom@maths.usyd.edu.au to let the CSOs know of any special requirements for the seminar.