The flexibility of caustics
Abstract/Contents
- Abstract
- In this thesis we establish a full h-principle for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we show that if the obvious homotopy theoretic obstruction to the simplification of singularities vanishes, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The h-principle is full in that it holds in C^0-close, relative and parametric versions. Among several applications of the h-principle we obtain a generalization of the Reidemeister theorem for Legendrian knots in the standard contact R^3, which allows for the simplification of the singularities of the front of a family of Legendrian knots parametrized by a space of arbitrary dimension. To prove our result we use two well-known tools in the philosophy of the h-principle: the holonomic approximation lemma and the wrinkled embeddings theorem. However, both of these tools need to be upgraded in order to be applicable to the situation at hand. For this purpose we refine the holonomic approximation lemma to a version in which cutoffs can be carefully controlled and we adapt the wrinkled embeddings theorem to the setting of Lagrangian and Legendrian embeddings.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2018 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Alvarez-Gavela, Daniel |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Eliashberg, Y, 1946- |
| Thesis advisor | Eliashberg, Y, 1946- |
| Thesis advisor | Ionel, Eleny |
| Thesis advisor | Starkston, Laura |
| Advisor | Ionel, Eleny |
| Advisor | Starkston, Laura |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Daniel Alvarez-Gavela. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2018. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2018 by Daniel Alvarez-Gavela
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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