Conlon will investigate the structure of codimension one foliations. The existence of foliations of knot complements, leafwise hyperbolic structures, smoothability of foliations, and ergodic properties will be studied. This work expands the Principal Investigator's theory of levels which has made a profound impact on development of foliation theory and has led to interaction with the French and Japanese schools of foliations. The mathematical theory of foliations concerns the filling of space by stacks of surfaces or "leaves". Conlon helped to prove that any surface may be a leaf of a foliation. Since some surfaces wrap upon themselves, this proof was very difficult. Now Conlon will further his theory by understanding foliations which fill all of a space except for a knot. The understanding of knots has recently become very important to the understanding of thermodynamics and molecular biology.
