Roman Golovko: Estimating the number of Reeb chords using linear representations

Given a chord-generic horizontally displaceable Legendrian submanifold L of P x R with the property that its characteristic algebra admits a finite-dimensional matrix representation, we show an Arnold-type lower bound for the number of Reeb chords on L. This result is a generalization of the results of Ekholm-Etnyre-Sullivan and Ekholm-Etnyre-Sabloff which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds L of C^n x R, n > 0, whose characteristic algebras admit finite-dimensional matrix representations, but whose Chekanov-Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct examples of Legendrian submanifolds L of C^n x R with the property that the characteristic algebra of L does not satisfy the rank property. This is a joint work with Georgios Dimitroglou Rizell.