This is a proposal to provide funding for the conference "Geometry and Topology in Samos" to be held in Samos, Greece during June 11-16, 2012. The focus of this conference is the classification of manifolds, including both topological and geometric aspects. The central rigidity conjecture is the Farrell-Jones Conjecture, which generalizes both the Novikov and Borel rigidity conjectures. The conference will bring together researchers who use geometric methods to study topological problems with researchers who use topological methods to study geometric problems. Examples of the former are high-dimensional topologists and examples of the latter are geometric group theorists and differential geometers.
Topology is often described as "rubber geometry": spaces are considered the same if one can be deformed into another via stretching and compressing (but without tearing or puncturing). A basic problem is to decide whether two spaces can be deformed into each other. The primary method for doing this is to develop ways to assign computable invariants to the space: if two spaces can be deformed into each other, then the associated invariants have to coincide. And conversely, if two spaces have different invariants, then they cannot be deformed into each other. An example of such invariants are the homotopy groups of a space. One can then ask whether the homotopy groups are good enough invariants to determine spaces. In other words, if if we have two spaces whose homotopy groups are the same, can they be deformed into each other? One of the central conjectures in topology is the Borel Conjecture, which predicts that for a certain class of spaces, the answer to the previous question is "yes". This problem has been central to high-dimensional topology, and work on it has involved sophisticated techniques from areas as diverse as algebra, geometry, and dynamics. The conference plans on bringing together international experts whose work touches upon the Borel conjecture (and related questions), with a view towards charting the course of future research on these topics.
", which was held in Samos, Greece, during the week of June 11th-15th, 2012. Geometry originated in ancient Greece, and one of the earliest practitioners, Pythagoras (of the famed a^2 + b^2 = c^2), lived on the island of Samos. Topology is a more recent development, having its roots in France in the 1800's. Over the course of the last 30 years, there have been a great deal of interactions between these two fields. The conference in Samos featured talks by 27 world class researchers (15 from the US), working at the interface of these two fields. The talks discussed the latest research developments in these areas. In addition, a public lecture was given on "A modern assessment of the legacy of Pythagoras". There were 61 total participants at the conference (17 from the US), including a large number of doctoral students working in geometry and topology. NSF funds were primarily used to help defray the transportation and hotel costs for US participants. In addition, financial assistance was provided to a few of the speakers who would otherwise have been unable to attend. Finally, some partial support was provided to the graduate students. During the conference, there was a great deal of discussion and activity between the participants. Several groups of participants initiated new research projects, and there have been followup collaborative visits between various participants. Funding of non-US graduate students helped build up "good will", and increased the likelihood that some of the strongest of these students will take up visiting positions in US institutions upon completion of their PhD.
