University of Toronto
                  PDE/Applied Math/Analysis Seminar
                    Wednesday 14 February, 3:10-4:00pm
			   Fields Institute



SPEAKER:  Adam Oberman

TITLE:    The Infinity Laplacian: from classical analysis to image 
	  processing and random turn games.

ABSTRACT: The Infinity Laplacian equation is currently at the interface 
	  of a number of different mathematical fields. It was first 
	  studied in the 1950s by the Swedish mathematician Gunnar 
	  Aronsson, motivated by classical analysis problem of building 
	  Lipschitz  extensions of a given function. While Aronsson was 
	  able to find interesting exact solutions, progress  stalled 
	  because solutions were non-classical. It took another forty 
	  years until analytical tools were developed to  study the 
	  equation rigorously, and computational tools were developed 
	  which made numerical solution  of the equation possible.

	  In the last decade, PDE theorists established existence and 
	  uniqueness, and (quite recently) appropriate regularity results. 
	  At the same time, the image processing community was using the 
	  operator for edge detection, and for inpainting, the reparation 
	  of images with damage. While the operator was promising, they 
	  had little success, since traditional methods for solving the 
	  equation yielded poor results.

	  It turns out that the right way to solve the equation is to go 
	  back to the original Lipschitz extension problem. This leads to 
	  a formula for the discrete operator with a simple interpreation, 
	  and good solution properties. This formula also leads to another 
	  surprising connection with probability theory.

	  Working in the unrelated field of percolation theory, a group 
	  of probalists (Peres-Shramm-Sheffield-Wilson) studying a 
	  randomized version of a marble game called Hex found a connection 
	  with the Infinity Laplacian equation. This connection gives an 
	  interpretation of the equation as a two player random game.

	  I'll tell this story, and explain some of the more accessible 
	  properties of the equation, along with pictures and numerical 
	  results.


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University of Toronto PDE/Applied Math/Analysis Seminar 
http://www.math.toronto.edu/appmath/

2006-2007 organizers:
          Pieter Blue           pblue@ math.toronto.edu
          Almut Burchard        almut@ math.toronto.edu
          Robert McCann		mccann@ math.toronto.edu
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