In the 50s, Smale and Hirsch proved the surprising fact that the space of immersions of a manifold into another manifold (of strictly greater dimension) is homotopy equivalent to the space of bundle monomorphisms between the tangent bundles. Effectively, this says that, despite superficial appearances, immersions are purely governed by homotopy theory.

In modern terms, we say that the immersion problem is flexible or, following Gromov, that it abides by the h-principle. Starting in the late 60s, Gromov showed that many other geometric problems exhibit flexibility. Many more h-principles have been discovered since, including in geometric topology, symplectic topology, complex geometry, Riemannian geometry, and beyond. To this day new flexibility phenomena are being discovered and applied, often with surprising consequences.

This summer school will include foundational introductory courses on some of the main h-principle techniques (holonomic approximation, convex integration, wrinkling) as well as on recent applications to geometry and topology.


Please register for the summer school here

The deadline for registration is February 29th


We will have some funding available for PhD students. Please indicate in the registration form whether you are requesting accommodation (which we would book for you) and whether you are requesting funding for travel. We strongly encourage applications from as diverse a pool of candidates as possible!