Birkhoff section is an important notion for 3-dimensional dynamical systems.
In 90', Hofer, Wysocki and Zehnder showed that there must exist a Birkhoff section of disk type binding a periodic orbit for the Reeb vector field in convex contact 3-sphere by using pseudoholomorphic curves coming from ellipsoids. It is a natural question whether the same result holds in lens spaces. Recently Reeb flows in convex lens spaces have been studied by Hryniewicz and Salomao and it was shown that a similar result holds in some lens spaces under certain conditions by developing the original Hofer, Wysocki and Zehnder's techniques.

On the other hand, there is a powerful tool for studying contact 3-manifolds called Embedded contact homology, which was constructed by Hutchings in 00' as a 3-dimensional analogues of Gromov invariant defined by Taubes in 4-symplectic manifolds.

In this talk, I will introduce an approach to the existence of a Birkhoff section of disk type in lens spaces using ECH and in addition, the connection between their contact areas and the first ECH spectrums.