The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f:(ℝ^{2n},σ = ∑_{i=1}^{n} dx_i ∧ dy_i) → (ℝ^{2n},σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2n} × ℝ^{2n},ω = π*₂σ - π*₁σ)$. This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $SΛ_{2n} ≃ SU(2n)/SO(2n)$ is defined.
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We study Thom polynomials associated with Lagrange singularities. We expand them in the basis of Q̃-functions. This basis plays a key role in the Schubert calculus of isotropic Grassmannians. We prove that the Q̃-function expansions of the Thom polynomials of Lagrange singularities always have nonnegative coefficients. This is an analog of a result on the Thom polynomials of mapping singularities and Schur S-functions, established formerly by the last two authors.
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