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We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

Given a compact Riemannian surface $M$, with Laplace-Beltrami operator $\Delta$, for $\lambda > 0$, let $P_{\lambda,\lambda^{-\frac{1}{3}}}$ be the spectral projector on the bandwidth $[\lambda-\lambda^{-\frac{1}{3}}, \lambda + \lambda^{\frac{1}{3}}]$ associated to $\sqrt{-\Delta}$. We prove a polynomial improvement on the $L^2 \to L^{\infty}$ norm of $P_{\lambda,\lambda^{-\frac{1}{3}}}$ for generic simple spheres of revolution (away from the poles and the equator) and for the Euclidean disk away from its center but up to the boundary. We use the Quantum Integrability of those surfaces to express the norm in terms of a joint basis of eigenfunctions for $\left(\sqrt{-\Delta}, \frac{1}{i}\frac{\partial}{\partial \theta}\right)$. Then, we use that those eigenfunctions are asymptotically Lagrangian oscillatory functions, each supported on a Lagrangian torus with fold-type caustic. Thus, studying the distribution of the caustics, and using BKW decay away from the caustics, we are able to reduce the problem to counting estimates.

Here are seemingly unrelated problems: computing rational homotopy groups of spheres in rational homotopy theory, purity in algebraic geometry, Koszul duality for the category of a reductive group in representation theory, splitting Drinfeld space's de Rham complex in the p-adic Langlands program, deformation quantization of Poisson manifolds in mathematical physics. And yet, all of them boil down to the same question: formality. A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, a Pre-Calabi-Yau algebra, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. This thesis develops obstruction classes allowing to prove formality results. On the one hand, it incorporates aforementioned results into a single theory. On the other hand, it provides tools to study these questions in cases little studied hitherto: over any coefficient ring and for algebraic structures with several outputs: algebras encoded by properads.