Seidel's Lagrangian Dehn twist exact sequence has been a
cornerstone of the theory of Fukaya categories. In the last decade,
Huybrechts and Thomas discovered a new autoequivalence in the derived
cateogry of coherent sheaves using the so-called "projective objects", which
are presumably mirrors of Lagrangian projective spaces. On the other hand,
Seidel's construction of Lagrangian Dehn twists as symplectomorphisms can be
easily generalized to Lagrangian projective spaces. The induce
auto-equivalence on Fukaya categories are conjectured to be the mirror of
Huybrechts-Thomas's auto-equivalence on B-side.
This remains open until recently, and I will explain my joint work with
Cheuk-Yu Mak on the solution to this conjecture using the technique of
Lagrangian cobordisms. Moreover, we will explain a recent progress, again
joint with Cheuk-Yu Mak, on pushing this further to Lagrangian embeddings of
finite quotients of rank-one symmetric spaces, leading to another new class
of auto-equivalences, which are different from the classical spherical
twists only in coefficients of finite characteristics.