Doctoral Thesis by

TLDR: In this thesis, the existence and uniqueness of gradient trajectories near an A2singularity are analysed and it is proved that the two Lagrangian vanishing cycles associated to these critical points intersect transversally in exactly one point in all regular fibres along a straight line.

Abstract: In this thesis, the existence and uniqueness of gradient trajectories near an A2singularity are analysed. The A2-singularity is called a birth-death critical point in the real case. The birth-death critical point appears in a one-parameter family of functions. Such a family of functions has precisely two Morse critical points of index difference one, on the birth side. The result of the real case states that these two critical points are joined by a unique gradient trajectory up to time-shift. Here the gradient flow is defined with respect to any family of Riemannian metrics. This can be viewed as a converse to Smale’s cancellation theorem. We also look at the complex analogue of the result in Picard–Lefschetz theory. This analogue considers a holomorphic one-parameter family with an A2-singularity. Such a family has two critical Morse critical points near the singularity for every small non-zero parameter. We prove that the two Lagrangian vanishing cycles associated to these critical points intersect transversally in exactly one point in all regular fibres along a straight line. The result is obtained by analysing the gradient trajectories of the real part of these functions. Both proofs start with a normal form in local coordinates for such families of functions. The gradient equations in these coordinates can be rescaled into a fast-slow system of non-linear differential equation. Existence will rely on an adiabatic limit analysis whereas uniqueness follows from a Conley index pair construction. The latter construction will also show that connecting gradient trajectories cannot leave the local charts. Even though the proof of these two results follow from similar lines of argument, the real case cannot be reduced to the complex case and vice versa.