Mathematical Sciences: Global Differential Geometry and Singularities

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The principal investigators will study problems involving singularities in algebraic and differential geometry. In particular they will consider the singularities of Gauss maps and Lagrangian structures, tangent cones of the Thom-Boardman singularity loci, and Stiefel -Whitney classes of singular real analytic varieties. The broad background of the six investigators will allow them to apply a range of methods which might not ordinarily be focused on this set of problems. In particular they will apply sheaf theoretic methods, flows on Jacobians, and integral geometry in their research. Singularities in geometry present a set of problems much more difficult than those in which the surfaces studied are smooth. Consequently much less is known about this case and more sophisticated techniques are needed to make progress. On the other hand our physical world is full of examples of spaces in which singularities are important - from fractures in materials to boundaries between liquid and solid phases of metals. Describing and understanding these singularities is important from both a practical and theoretical standpoint. The principal investigators will use their varied backgrounds to study this important collection of problems.