By using generating functions, Viterbo constructed spectral invariants for Lagrangians of cotangent bundles in 1992. Ten years later, Oh and Schwarz (independently) adapted the construction to Hamiltonian diffeomorphism groups of quite general manifolds thanks to Floer theory. Since then, spectral invariants were defined in various contexts and yielded a great number of applications of different natures.
I will tak about joint work with Frol Zapolsky in which we define spectral invariants for monotone Lagrangians and establish the properties which make them such a useful tool. In particular, given a monotone Lagrangian L, I will show how spectral invariants can be seen as functions defined on the universal cover of the product of the set of Lagrangians Hamiltonian isotopic to L and the Hamiltonian diffeomorphism group. Then, I'll show that these functions are Lipschitz with respect to the natural Hofer distance on this space.