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
