Michael Albanese, Department of Pure Mathematics, University of Waterloo
"The Hitchin-Thorpe Inequality"
Every two-dimensional manifold admits a metric of constant curvature by the Uniformisation theorem. In higher dimensions, one could try to generalise this fact by asking for the existence of a metric of constant scalar curvature, constant Ricci curvature, or constant sectional curvature (these all coincide in dimension two). Each condition is more restrictive than the last. There is a plethora of constant scalar curvature metrics on every manifold, while constant sectional curvature metrics rarely exist, so focus often turns to metrics of constant Ricci curvature, also known as Einstein metrics. In dimensions five and above, there are no known examples of manifolds which fail to admit such a metric. This is in stark contrast to dimension four where the Hitchin-Thorpe inequality provides a topological obstruction. We will give the proof of this inequality, following Hitchin's original paper "Compact four-dimensional Einstein manifolds".