Michael Albanese, Department of Pure Mathematics, University of Waterloo
"Almost Complex Four-Manifolds with no Complex Structure"
Which manifolds admit a complex structure? This is a very difficult problem with no concrete answer in general. To make some progress, it is useful to consider an intermediate structure known as an almost complex structure, whose existence is much easier to detect. Every complex manifold has an almost complex structure, so one is naturally led to consider the converse: does a manifold which admits an almost complex structure also admit a complex structure? This question has a positive answer in dimension two, a negative answer in dimension four, and is completely open in higher dimensions. We will discuss the results related to these conclusions and focus on why four dimensions is special.