# Textes du projet Microlocal

#### A short review on microlocal sheaf theory par Pierre Schapira

This is a brief survey of the microlocal theory of sheaves of Sheaves on manifolds, with some complements and variants.

#### The three cusps conjecture par Stéphane Guillermou

We prove Arnol'd's three cusps conjecture about the front of Legendrian curves in the projectivized cotangent bundle of the 2-sphere. We use the microlocal theory of sheaves of Kashiwara and Schapira and study the derived category of sheaves on the 2-sphere with a given smooth Lagrangian microsupport.

#### A lemma for microlocal sheaf theory in the ∞-categorical setting par Pierre Schapira et Marco Robalo

Microlocal sheaf theory of \cite{KS90} makes an essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in ℝ with values in the category Sets of sets be constant. In a first part of this paper, using classical tools, we show how to generalize the extension lemma to the case of the unbounded derived category. In a second part, we extend Kashiwara's result on constant functors by replacing the category Sets with the ∞-category of spaces and apply it to generalize the extension lemma to ∞-sheaves, the ∞-categorical version of sheaves. Finally, we define the micro-support of sheaves with values in a stable (∞,1)-category.

#### Microlocal analysis and beyond par Pierre Schapira

We shall explain how the idea of microlocal analysis of the seventies has been reformulated in the framework of sheaf theory in the eighties and then applied to various branches of mathematics, such as linear partial differential equations or symplectic topology.