I will present recent results on microlocal sheaves of spectra supported on a smooth Lagrangian in a cotangent bundle, and some of the applications. I will discuss the ideas and techniques of the proofs, including the use of an $(\infty,2)$-category of correspondences, and results from sheaf quantizations.

For an exact Lagrangian $L$ in a cotangent bundle, one can define a sheaf of stable-infinity categories on it, called the Kashiwara--Schapira stack. Assuming the Lagrangian is smooth, the sheaf of categories is a local system with fiber equivalent to Mod(k), where k is the coefficient ring spectrum (at least E_2). I will show that the classifying map for the local system of categories factors through the stable Gauss map L--->U/O and the delooping of the J-homomorphism U/O-->BPic(S), where S is the sphere spectrum. Part of the proof employs the (\infty,2)-category of correspondences developed by Gaitsgory--Rozenblyum. If time permits, I will also talk about some applications of this result.