Algebraic and geometric methods in four dimensions

This workshop will focus on the geometric and algebraic topology of manifolds. The qualitative behaviour of manifolds varies significantly with their dimension. In a sense, dimension four is the boundary case between low and high dimensions; there is enough freedom for the manifold topology to exhibit complex behaviour, but not enough room for the usual high dimensional tools to work. In particular, there is an astonishing divergence between the smooth and topological settings for 4-manifolds. Moreover, deep questions such as the Poincare conjecture and the Schoenflies conjecture remain open in dimension four.

Recent years have seen a resurgence of interest in 4-dimensional manifolds, both on the topological side using techniques pioneered by Freedman and Quinn, as well as on the smooth side due to Gabai’s 4-dimensional light bulb theorem and Watanabe’s disproof of the 4-dimensional Smale conjecture. Notably this progress can be seen as the result of inspiration drawn both from research on 3-manifolds, including knots and links, as well as on higher-dimensional manifolds. With a new generation of researchers focusing their interest on this area, the time is ripe for new developments.

The aim of this workshop is to bring together junior and senior researchers in 4-manifolds, as well as the neighbouring fields of 3-manifolds and higher-dimensional manifolds, stimulating further collaboration on fundamental questions. In particular, we hope to strengthen the ties between the Australian and international research communities.

This MATRIX Research Program is partially supported by AMSI and AustMS through the AMSI-AustMS Workshop Funding program, and benefits from the SMRI International Visitor Program.