Mathematical Sciences: Topology, Geometry, and Dynamics of Foliations

Conlon将研究余维叶的结构。结补叶理的存在性、叶面双曲结构、叶理的平滑性和遍历性将被研究。这项工作扩展了首席研究员的水平理论，该理论对叶理理论的发展产生了深远的影响，并导致了与法国和日本叶理学派的互动。叶的数学理论关注的是由成堆的表面或“叶子”来填充空间。康伦帮助证明了任何表面都可能是一个叶状结构的叶子。因为有些表面是自己包裹起来的，所以这个证明是非常困难的。现在Conlon将通过理解叶状结构来进一步深化他的理论，叶状结构填满了除了结以外的所有空间。最近，对结的理解对热力学和分子生物学的理解变得非常重要。

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.