  This is a lecture note from a seminar course given at Tel Aviv University in
Spring 2018. Part of the preliminary section is built from Kazhdan's seminar
organized in the Hebrew University of Jerusalem in Fall 2017. The main topic of
this note is a detailed introduction of Tamarkin category theory from a
quantitative perspective, followed by a demonstration of various applications
in symplectic topology. Many examples are provided in order to obtain certain
geometric intuitions of those abstract algebraic constructions in Tamarkin
category. In this note, we try to explain how standard symplectic techniques,
for instance, generating function, capacities, symplectic homology, etc., are
elegantly packaged in the language of sheaves as well as related intriguing
sheaf operators. In addition, many concepts developed in Tamarkin category
theory are natural generalizations of persistent homology theory, so their
relations are emphasized along with the development of the note.
