Compound Du Val singularities, five-dimensional SCFTs and GV invariants

In this thesis work we introduce a new method to study the dynamics of M-theory on compound Du Val threefold singularities (cDV). Incidentally, this also furnishes a new way to systematically count the Gopakumar-Vafa invariants (GV) of these geometries and, reversely, to produce threefolds whose GV invariants display required properties. Our construction is inspired by the type IIA limit of M-theory on the considered singularities and rephrases the data of the threefolds in the language of seven-dimensional super Yang-Mills theory. This, more deeply, creates a connection between the algebraic properties of the ADE algebras and the geometric properties of the cDVs. We focused our analysis on two interesting classes of compound Du Val: the simple flops and the quasihomogeneous cDVs, obtaining in both cases complete information on the GV invariants (or, equivalently, on the Higgs Branch of M-theory reduced on these singularities). We also elucidate, during this procedure, the role of exotic type IIA branes bound states, called T-branes, that lack a clear interpretation in terms of the geometry of the threefold.