** Title: **
On the homotopy type of nearby Lagrangians

**Abstract:**
Fukaya-Seidel-Smith and Nadler proved that the inclusion of an exact
Lagrangian of Maslov index 0 in a simply connected cotangent bundle induces an
isomorphism on homology. I will show that it is in fact a homotopy equivalence,
dropping the assumption on the fundamental group. There are three new ideas
required to prove this: (1) the fact that a cotangent fibre generates the
(wrapped) Fukaya category of a cotangent bundle, (2) considering a Fukaya
category in which Lagrangians equipped with local systems of arbitrary rank are
allowed as objects and, (3) defining a Fukaya category associated to the
universal cover of a compact manifold. I will illustrate (2) and (3) with
elementary examples (e.g. coming from Morse theory) to illustrate why a new idea
is needed.