Mark Powell studies the topology of manifolds, especially in dimensions three and four, together with homological algebra. His Glasgow profile verifies this research specialization.[ref]
https://www.gla.ac.uk/schools/mathematicsstatistics/staff/markpowell/. Institutional research statement
His Introduction to Differential Geometry by Mark Powell connects the local geometry of curves and surfaces with global invariants. Curvature and torsion of a curve describe how a curve bends in space, whereas the Gauss-Bonnet theorem turns integrated surface curvature and boundary terms into an Euler characteristic. These examples show how local differential measurements can constrain a whole geometric object.[ref]
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