Analysis of singularities in geometry

Geometric Analysis, and in particular Geometric Flows, led to some of the most important breakthroughs in pure mathematics in the last decades. This modern area of mathematics combines Differential Geometry (algebraic structures of geometric objects and local control of geometric properties such as curvature) with tools from Analysis (mainly partial differential equations and the calculus of variations) to obtain global differential topological results. As the partial differential equations that are involved are generally non-linear, they often give rise to singularities. In many interesting problems, it is important to understand when and how these singularities occur and to study precise characterisations of them.

An illustrative example is the Mean Curvature Flow. This flow evolves an initial manifold towards an object with more symmetries, for example a potato-shaped surface is transformed to a perfectly round sphere. In more general situations however, the flow develops local singularities where the curvature becomes infinitely large while staying bounded in other regions. Another example is the Ricci Flow, which can be seen as the intrinsic sibling of Mean Curvature Flow, used by Perelman to resolve the Poincaré and Geometrization conjectures, for which he won a Clay Millennium Prize and a Fields Medal.