Legendrian intersections in twisted 1-jet bundles

Author

• Alfredo Vargas Jr., Bryn Mawr College

Degree Date

2026

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Abstract

The Arnold conjecture for Lagrangian intersections in the setting of cotangent bundles with their standard symplectic structure has long been settled, as has its contact analog for Legendrian intersections in the setting of 1-jet bundles with their standard contact structure. This latter result was due originally to Chekanov, and was later refined by Bhupal. More recently, Chantraine and Murphy proved an analog of the Arnold conjecture for Lagrangian intersections in cotangent bundles with respect to their standard LCS (locally conformally symplectic) structures. This dissertation establishes a refinement of Chantraine and Murphy’s theorem analogous to Bhupal’s refinement of Chekanov’s theorem, where instead of the standard contact structure on J¹M we work with respect to the “twisted” contact structure lifting a given standard LCS structure on the underlying cotangent bundle. The proof involves defining an extension of the classical action functional in order to establish a variational principle for contact isotopies in the twisted sense, then carefully leveraging this to obtain special finite dimensional generating families, with explicitly calculable Morse index, for the associated Legendrians.

Citation

Vargas Jr., Alfredo. 2026. "Legendrian intersections in twisted 1-jet bundles." PhD Diss, Bryn Mawr College.