Ana Cannas da Silva, Instituto Superior Técnico
Fold-Forms on Four-Folds

A differential problem is said to satisfy the h-principle if any formal solution (i.e., a solution for the associated algebraic problem) is homotopic to a genuine (i.e., differential) solution. Gromov built the machinery of the h-principle on the work of Whitney, Nash, Kuiper, Smale, Hirsch, Poénaru, Phillips, etc. Without assuming familiarity with the h-principle, I will explain its application to prove the existence, on any orientable 4-manifold, of a closed 2-form with only fold type singularities. A formal solution necessary to conclude the argument is a stable complex structure, guaranteed by a result of Hirzebruch and Hopf.