Topology of Manifolds: Interactions between High and Low Dimensions

Indiana University, Bloomington IN

Investigators

Abstract

This award supports funding for the participation of US-based participants in the program "Topology of Manifolds: interactions between high and low dimensions" to be held January 7-18, 2019 at the University of Melbourne, Creswick, Australia. This meeting will bring together students, postdocs, and researchers from all over the world to stimulate research on fundamental questions in manifold theory. It will promote the interaction between researchers in high and low dimensional topology. The meeting is structured as follows: mini-courses in the first week by world experts and a conference in the second week. Both weeks focus on open problems and collaborative work. This structure will greatly benefit early career researchers. Another feature of the meeting that will make it accessible is the theme of the program: promoting interactions high and low dimensions will mitigate the tendency of technical talks and problems. There are two main research aims for this meeting. The first is to identify settings for synergy from the interaction between high and low dimensions and to make progress on problems in these settings. The second is to produce a high-quality problem list to guide future research in manifold topology. It is expected that a well-crafted and publicized problem list arising from the collaboration during the meeting will be of long-term benefit to the mathematical community. An n-manifold is a space which locally resembles n-dimensional Euclidean space. Manifolds of dimension less or equal than 3 are studied using geometric techniques. Manifolds of dimension greater or equal than 5 are studied via surgery theory, which involves a mix of algebraic and differential topology, algebra, and analysis. Dimension 4 is in between; both the high dimensional Whitney Trick and the low-dimensional geometric techniques are only partially successful. The need for the program is that these areas have diverged in the last several decades, to the extent that, often researchers in low/middle/high dimensional topology are not always aware of the current research/techniques in other dimensions. This program is expected to lead to a synergy, benefiting both the experts and the new generation of early career researchers. The website for the event can be found at https://www.matrix-inst.org.au/events/interactions-between-topology-in-high-and-low-dimensions/ This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.