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ETH Zürich

1. Antony, Charel. Gradient Trajectories Near Real And Complex A2-singularities.

Degree: 2018, ETH Zürich

URL: http://hdl.handle.net/20.500.11850/284182

In this thesis, the existence and uniqueness of gradient trajectories near an A_{2}-singularity are analysed. The A_{2}-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 PicardÂ â€“ Lefschetz theory. This analogue considers a holomorphic one-parameter family with an A_{2}-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.
*Advisors/Committee Members: Salamon, Dietmar, Frauenfelder, Urs, Biran, Paul.*

Subjects/Keywords: Birth-death; Critical point; Gradient flow; A_2 singularity; vanishing cycles; Whitney Lemma; Adiabatic Limit; Conley Index Pair; Existence and uniqueness of solutions; info:eu-repo/classification/ddc/510; Mathematics

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APA (6^{th} Edition):

Antony, C. (2018). Gradient Trajectories Near Real And Complex A2-singularities. (Doctoral Dissertation). ETH Zürich. Retrieved from http://hdl.handle.net/20.500.11850/284182

Chicago Manual of Style (16^{th} Edition):

Antony, Charel. “Gradient Trajectories Near Real And Complex A2-singularities.” 2018. Doctoral Dissertation, ETH Zürich. Accessed April 16, 2021. http://hdl.handle.net/20.500.11850/284182.

MLA Handbook (7^{th} Edition):

Antony, Charel. “Gradient Trajectories Near Real And Complex A2-singularities.” 2018. Web. 16 Apr 2021.

Vancouver:

Antony C. Gradient Trajectories Near Real And Complex A2-singularities. [Internet] [Doctoral dissertation]. ETH Zürich; 2018. [cited 2021 Apr 16]. Available from: http://hdl.handle.net/20.500.11850/284182.

Council of Science Editors:

Antony C. Gradient Trajectories Near Real And Complex A2-singularities. [Doctoral Dissertation]. ETH Zürich; 2018. Available from: http://hdl.handle.net/20.500.11850/284182