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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.