Lagrangian cobordism and tropical curves

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.

Contact round surgery and Lutz twists

From the viewpoint of contact round surgery, the Lutz twist and the Giroux torsion on contact [Formula: see text]-manifolds are studied in this paper. It is proved that the Lutz twist along a transverse knot and that along a pre-Lagrangian torus are realized by contact round surgeries of index [Formula: see text] and [Formula: see text] of contact [Formula: see text]-manifolds. This construction gives an important perspective on the Lutz twists. Both can be regarded as fiberwise connected sums with the compactified Lutz twist elements.