Dendroidal simplicial sets satisfying an analogue of the Segal condition are a model for ($\infty$, 1)-colored operads, as shown by Cisinski and Moerdijk. We consider weak group actions on such a ``Segal operad'' and prove a rigidification theorem. This is joint work with Julie Bergner.

We investigate non-degenerate Lagrangians of the form $$ \int f(u_x, u_y, u_t) dx dy dt $$ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the so-called method of hydrodynamic reductions. The integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the `master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball.

Quillen’s derived functor notion of homology provides interesting and useful invariants in a wide variety of homotopical algebraic contexts. For instance, in Haynes Miller’s proof of the Sullivan conjecture on maps from classifying spaces, Quillen homology of commutative algebras (Andre-Quillen homology) is a critical ingredient. Working in the topological context of structured ring spectra, this talk will introduce several recent results on localization and completion with respect to topological Quillen homology of commutative ring spectra (topological Andre-Quillen homology), $E_n$ ring spectra, and operad algebras in spectra. This includes homotopical analysis of a completion construction and strong convergence of its associated homotopy spectral sequence. The localization and completion constructions for structured ring spectra are precisely analogous to Sullivan's localization and completion of spaces (for which he recently won the Wolf prize), and Bousfield-Kan's version of Sullivan's localization and completion called the $R$-completion of a space with respect to a ring $R$. This is joint work with Michael Ching.