Reticular Lagrangian singularities

Lagrangian and Legendrian Singularities

These are notes of the mini-courses we lectured in Trieste in 2003 and Luminy in 2004. The courses were based on the books [1, 3, 2]. A symplectic form ω on a manifold M is a closed 2-form, non-degenerate as a skew-symmetric bilinear form on the tangent space at each point. So dω = 0 and ω n is a volume form, dim M = 2n. Manifold M equipped with a symplectic form is called symplectic. It is nec...

Finite-time Euler singularities: A Lagrangian perspective

Rigid and Complete Intersection Lagrangian Singularities

In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in [SvS03]. The result can be applied to show the rigidity of all open swallowtails of dimension ≥ 2. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show tha...

Singularities of Lagrangian Mean Curvature Flow: Zero-maslov Class Case

We study singularities of Lagrangian mean curvature flow in C when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonian isotopic to a plane that, nevertheless, develop finite time singularities under ...

Stability Criteria and Classification of Singularities for Equivariant Lagrangian Subwianifolds

One of the useful methods of mathematical physics is the one arising from symplectic geometry and associating the singularities of Lagrangian submanifolds with the optical caustics, phase transitions, bifurcation patterns, obstacle geometry etc. In this paper we derive the stability criteria for singularities of rquivariant Lagrangian submanifolds with a compact Lie group action determined by a...