The study of random shapes started over 100 years ago as a collection of examples, e.g. those arising from Brownian Motion. It has turned out to be a meeting place for probability theory, mathematics, physics, combinatorics, computer science, and certain areas of algebra. Recent advances in areas diverse as brain imaging, astrophysics, nanotechnology, and communications and sensor networks have been driven by notions related to random shapes or motions, and random transport. The past decade has seen both an explosion of results as well as new structures (for example, O. Schramm’s SLE processes) that unify various problems. While much progress has been made, this is still a very young field. For example, one is lacking a theory similar to SLE for generating random surfaces. The purpose of this program is to bring together experts from these rapidly developing areas in mathematics and the sciences to share new ideas and study new problems. We are mainly concerned with structures in two or three dimensions, as they have a strong connection to biology and physics, but some of the topics to be covered concern higher dimensional Euclidean spaces and some problems with networks may have no specified ambient dimension. Besides these general areas, there will also be activity in the study of random shapes and complex geometries arising in brain mapping and astrophysics. The topics to be covered include:
Brownian and fractional Brownian Motion; SLE and related Löwner evolution; geometry of the Gaussian free field; self-avoiding random walk; percolation; random shapes and Wiener space in representation theory; random curves, surfaces, and growth processes; random minimal surfaces; random conformal or quasiconformal mappings; random Teichmüller theory and univalent dynamics; random welding maps; random triangulations and metrics on surfaces; 3D image processing for complex geometries; PDE’s related to growth processes.
Random curves and surfaces in conformal field theory, quantum gravity, and string theory; simulations of random curves and surfaces; folding, shrinking, wrinkling, and buckling of surfaces and membranes; random folding of polymers; geometry of random fields; electrodeposition and rough boundaries in electrochemistry; diffusion limited aggregation; branching structures and random transport; large scale cosmic fields and structures; random structures and diffusion in nanotechnology.
Random trees, circuits, graphs, branching processes, and related algorithms; random partitions and metrics; random polytopes; random routing and transport; random search algorithms; dynamic networks, graphs, and spanners; complex geometries in communication and sensor networks; 3D graphics for complex surfaces; computational geometry for random surfaces and sets.
Peter Jones, Chair (Yale University)
Richard Kenyon (University of British Columbia)
Stanley Osher (IPAM)
Nicholas Read (Yale University)
Steffen Rohde (University of Washington)
Bernard Sapoval (École Polytechnique)
Leon Takhtajan (SUNY Stony Brook)