##
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.