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In this talk we will review the notion of an h-principle, providing classic and modern examples, and emphasizing the main techniques available to prove them. Intuitively, an h-principle is a result which states that a geometric classification (e.g. of immersions, Riemannian metrics, symplectic or contact structures) can be achieved entirely by algebraic computations (e.g. of homotopy and cohomology groups). We will present the classical h-principles, due to H. Whitney, J. Nash and M. Gromov, and continue to more recent developments, such as the theory developed by Eliashberg-Mishachev and a new h-principle for codimension-2 contact embeddings. The purpose of the talk is that the audience gets a good sense of what h-principles are, how they work and how one proves them. The precise definition of an "h-principle", in terms of scanning maps, is admittedly rather convoluted, so we shall not today attempt further to define the kinds of material we understand to be embraced within that shorthand description, but you will know it when you see it.

This will be an in-person talk.