230525 Álvaro del Pino Gomez

One of the first highlights in the theory of manifolds is the result (1936) of Whitney stating that every smooth manifold can be embedded in a sufficiently large euclidean space. This is a quantitative result: Whitney proved that every n-dimensional manifold can be embedded in (2n+1)-dimensional space. He later introduced the Whitney trick (1944) and showed that 2n is sufficient.

This became a driving question during this period: what can be said about the spaces of embeddings/immersions between two given manifolds? Whitney himself provided in 1937 a complete description (up to homotopy) of the immersions of the circle into the plane. Twenty years later, Smale developed the method of corrugations to prove the sphere eversion theorem and, together with Hirsch, provided a complete classification (in terms of bundle monomorphisms) for immersions of subcritical dimension.

In 1969, Gromov turned the method of corrugations into a general machine (applicable in the study of many different geometric structures on manifolds) called the method of flexible sheaves, initiating the modern theory of (geometric) h-principle.

My aim with this talk is to provide a historical overview of these ideas, introduce the field of h-principle, and highlight some of the (many!) open questions in the area.