Program

In recent years probability theory (and here we mean probability theory in the largest sense, comprising combinatorics, statistical mechanics, algorithms, simulation) has made immense progress in understanding the basic two-dimensional models of statistical mechanics and random surfaces. Prior to the 1990s the major interests and achievements of probability theory were (with some exceptions for dimensions 4 or more) with respect to one-dimensional objects: Brownian motion and stochastic processes, random trees, and the like. Inspired by work of physicists in the ’70s and ’80s on conformal invariance and field theories in two dimensions, a number of leading probabilists and combinatorialists began thinking about spatial process in two dimensions: percolation, polymers, dimer models, Ising models. Major breakthroughs by Kenyon, Schramm, Lawler, Werner, Smirnov, Sheffield, and others led to a rigorous underpinning of conformal invariance in two-dimensional systems and paved the way for a new era of “two-dimensional” probability theory. Bibliography (PDF) Open Problems: 1- Aldous.pdf 2-Guttmann.pdf 3-Guttmann2.pdf 4-Kenyon.pdf 5-Linusson.pdf 6-Mossel.pdf 7-Rohde.pdf 8-Soteros.pdf 9-Winkler.pdf 10-Gorin.pdf 11-Wilson.pdf 12-Propp2.pdf 13-Difrancesco.pdf 14-Randall.pdf 15-Randall2.pdf 16-Young.pdf 17-Levine.pdf 18-Propp3.pdf 19-Propp4.pdf The problem Guttmann2.pdf above has been solved.  The solution is given in the appendix of the paper THE CRITICAL FUGALITY FOR SURFACE ADSORPTION OF SELF-AVOIDING WALKS ON THE HONEYCOMB LATTICE IS 1 + \sqrt{2} By Nicholas R. Beaton, Mireille Bousquet-Melou, Jan de Gier, Hugo Duminil-Copin, Anthony J. Guttman. arXiv:1109:0358v3. The problem appears as Theorem 10 in that paper, and the solution is given in the Appendix. (The paper will appear in Communications in Mathematical Physics.)