2014 Yamabe Memorial Symposium, September 17-19, 2014

研讨会将于2014年9月17日至19日在明尼苏达大学校园举行。它将集中在双曲3-流形的几何，格罗莫夫-双曲群的几何，以及其他相关的主题。在过去的40年左右的时间里，有一系列的发现彻底改变了对“大多数”三维流形的研究，将研究从拓扑学的主题变成了双曲几何，拓扑学和几何群论都是必不可少的组成部分。因此，这些流形可以从几何上进行研究，而且比单独使用拓扑学要深入得多。最近的工作使早期的工作更加深入，使之更加复杂。这项工作的发现者获得了2012年的克莱研究奖和2013年的Veblen奖，确立了早期研究人员猜测的双曲3-流形的性质。新的结果证实了双曲3-流形实际上具有表面子群定理、虚Haken定理和虚纤维定理所描述的性质。这个简短而紧张的研讨会的目的是向与会者介绍这项新工作的细节，并探索将新发现应用到更远的地方的选择方案。在数学中，寻求理解所有可能的三维形状的几何图形的历史由来已久。近年来，已经发现了新的技术来更深入地研究各种可能性。事实上，人们已经发现，大多数形状都有一个内部几何图形，称为双曲几何图形。这种几何的存在使数学家能够处理一般的形状，就像欧几里得几何被用来探索我们的三维世界一样。它产生了广泛而详细的几何描述，这在以前是不可能的。专题讨论会将讨论这项工作的技术细节以及进一步应用这项工作的可能性。这一奖项将支持大约40名研究生、博士后和初级教师参加研讨会。详情请浏览www.math.umn.edu/yamabe/2014

The Symposium will be held September 17-19, 2014, on the campus of the University of Minnesota. It will be centered on the geometry of hyperbolic 3-manifolds, the geometry of Gromov-hyperbolic groups, and additional allied topics. Within the past 40 years or so there has been a sequence of discoveries which have revolutionized the study of "most" three dimensional manifolds, turning the study from a subject in topology to one of hyperbolic geometry with both topology and geometric group theory being essential ingredients. As a result, these manifolds can be studied geometrically, and in much greater depth than with the use of topology alone. Recent work has compounded the earlier work by bringing even greater depth to it. This work, whose discoverers have received the 2012 Clay Research Price and the 2013 Veblen prize, established properties of hyperbolic 3-manifolds conjectured by earlier researchers. The new results confirmed that hyperbolic 3-manifolds in fact have the properties described in the Surface Subgroup Theorem, the Virtual Haken Theorem, and the Virtual Fibering Theorem. The purpose of this short, intense symposium is to bring to the participants details of this new work, and to explore options for applying the new discoveries even further afield. There is a long history in mathematics of seeking to understand the geometry of all possible three dimensional shapes. In recent years, new techniques have been discovered to delve much more deeply in the range of possibilities. Indeed, it has been discovered that most of the shapes have an internal geometry, called hyperbolic geometry. The existence of this geometry allows mathematicians to work with general shapes, much as Euclidean geometry is used to explore our three dimensional world. It has resulted in an extensive detailed geometric description that formerly had not been thought possible. The symposium will be concerned with discussing technical details of this work together with possible further applications of it. This award will support the participation of approximately forty graduate students, postdocs, and junior faculty in the symposium. Details are at www.math.umn.edu/yamabe/2014