After a general introduction recalling necessary definitions and notions to kickstart the workshop, we will present a work in progress that is aimed at giving lower bounds on the number of 1-periodic orbits of a class of Hamiltonians on blown-up LCS-ifications of hypertight contact manifolds in terms of Novikov homology, using Hamiltonian Floer theory.
Using generating functions quadratic at infinity for Lagrangian submanifolds of twisted cotangent bundles, we define spectral selectors for compactly supported lcs Hamiltonian diffeomorphisms of the locally conformal symplectizations \( S^1 \times \mathbb{R}^{2n+1} \) and \( S^1 \times \mathbb{R}^{2n} \times S^1 \) of \( \mathbb{R}^{2n+1} \) and \( \mathbb{R}^{2n} \times S^1 \), and obtain several applications: the construction of a bi-invariant partial order on the group of compactly supported lcs Hamiltonian diffeomorphisms of \( S^1 \times \mathbb{R}^{2n+1} \) and \( S^1 \times \mathbb{R}^{2n} \times S^1 \), of an integer-valued bi-invariant metric on the group of compactly supported lcs Hamiltonian diffeomorphisms of \( S^1 \times \mathbb{R}^{2n} \times S^1 \), and of an integer-valued lcs capacity for domains of \( S^1 \times \mathbb{R}^{2n} \times S^1 \). The lcs capacity is used to prove a lcs non-squeezing theorem in \( S^1 \times \mathbb{R}^{2n} \times S^1 \) analogous to the contact non-squeezing theorem in \( \mathbb{R}^{2n} \times S^1 \) discovered in 2006 by Eliashberg, Kim and Polterovich. Along the way we introduce for Liouville lcs manifolds the notions of essential Lee chords between exact Lagrangian submanifolds and of essential translated points of exact lcs diffeomorphisms. We prove that essential translated points always exist for time-\(1\) maps of sufficiently \(\mathcal{C}^0\)-small lcs Hamiltonian isotopies of compact Liouville lcs manifolds and for all compactly supported lcs Hamiltonian diffeomorphisms of \( S^1 \times \mathbb{R}^{2n+1} \) and \( S^1 \times \mathbb{R}^{2n} \times S^1 \). We also obtain an existence result for essential Lee chords between the zero section of a twisted cotangent bundle with compact base and its image by any lcs Hamiltonian isotopy, which can be thought of as a lcs analogue of the Lagrangian and Legendrian Arnold conjectures on usual cotangent and \(1\)-jet bundles. Finally, we introduce the notion of orderability for lcs manifolds, and prove that \( S^1 \times \mathbb{R}^{2n+1} \), \( S^1 \times \mathbb{R}^{2n} \times S^1 \) and twisted cotangent bundles are orderable.
In dimensions greater than four, the classification of smooth manifolds is an unsolvable problem, but manifolds can still be classified up to cobordism. From this perspective, Liouville cobordisms provide a powerful tool for studying contact manifolds in high dimensions. In this talk, I will explain how Liouville cobordisms can be used to construct exact locally conformally symplectic (LCS) manifolds, in particular the LCS mapping tori associated with a contactomorphism. I will then use this construction to study the isomorphism classes of LCS mapping tori and explore their connections with the contact mapping class group.