Complexes in low-dimensional topology

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。约翰·亨佩尔在《从曲线复形看3-流形》一书中阐述了曲线复形作为理解3-流形的工具的效用。他的工作将几何群论中使用的技术引入到低维流形的研究中。由Howard Masur和Yair Minsky建立的曲线复合体的Gromov双曲性提供了对复合体的结构和它所编码的信息的深刻见解。在这方面，侯赛因·纳马齐、索尔·施莱默和胡安·索托的狂热作品特别值得一提。近年来，P.I.研究了几个复合体，特别是宽度复合体(由P.I.定义)。和Kakimizu建筑群。这些复合体揭示了P.I.S钟爱的许多关于Heegaard分裂的亏格、阶和稳定性的问题。P.I.致力于低维拓扑领域的研究，特别是3-流形。该项目建立在低维拓扑学和几何群论的相互促进的基础上。P.I.与美国、德国和以色列的数学家保持着几个国内和国际合作。P.I.的目标是让研究生和本科生参与该项目的某些方面。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In "3-manifolds as viewed from the curve complex" John Hempel illustrated the utility of the curve complex as a tool for understanding 3-manifolds.His work introduced the techniques used in geometric group theory into the study of low-dimensional manifolds. The Gromov hyperbolicity of the curve complex, established by Howard Masur and Yair Minsky, provides deep insights into the structure of the complex and the information it encodes. The fanstastic work of Hossein Namazi, Saul Schleimer and Juan Souto deserves particular mention in this context. In recent years, the P.I. has studied several complexes, in particular the width complex (defined by the P.I.) and the Kakimizu complex. These complexes shed light on many of the P.I.'s pet problems on genus, rank and stabilization of Heegaard splittings.The P.I. pursues research in the area of low-dimensional topology, specifically 3-manifolds. The project builds on the cross-fertilization of low-dimensional topology and geometric group theory. The P.I. maintains several national and international collaborations with mathematicians in the United States, Germany and Israel. The P.I. aims to involve graduate and undergraduate students in some aspects of the project.