On Lagrangian Quasi-Cobordisms

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a preorder, and that they are not symmetric, it is not known if they form a partial order on Legendrian knots. The idea of a Lagrangian quasi-cobordism was first defined by Sabloff, Vela-Vick, and Wong. Roughly, for two Legendrians of the same rotation number, it is the smooth composition of a sequence of alternatingly ascending and descending Lagrangian cobordisms which start at one knot and ends at the other. This forms a metric monoid on Legendrian knots, with distance given by the minimal genus between any two Legendrian knots. In this talk, I will discuss some new results about Lagrangian quasi-cobordisms, based on some work in progress with Sabloff, Vela-Vick, and Wong.