Abstract:
This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation covariant derivatives for a number of interesting examples in projective, conformal and Grassmannian geometries.

Abstract:
The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) is explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on linear sections of the Lagrangian Grassmannian Sp(3)/U(3) with orthogonal linear sections of the dual variety and vice versa. A main technical result of independent interest is the fact that any nodal hyperplane section of the Lagrangian Grassmannian projected from the node is a linear section of the Grassmannian Gr(2,6).

Abstract:
In this article, we show that the Fredholm Lagrangian Grassmannian is homotopy equivalent with the space of compact perturbations of a fixed lagrangian. As a corollary, we obtain that the Maslov index with respect to a lagrangian is a isomorphism between the fundamental group of the Fredholm Lagrangian Grassmannian and the integers.

Abstract:
The stratification of the Grassmannian by positroid varieties has been the subject of extensive research. Positroid varieties are in bijection with a number of combinatorial objects, including $k$-Bruhat intervals and bounded affine permutations. In addition, Postnikov's boundary measurement map gives a family of parametrizations of each positroid variety; the domain of each parametrization is the space of edge weights of a weighted planar network. In this paper, we generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian $\Lambda(2n)$, which is the type $C$ analog of the ordinary, or type $A$, Grassmannian. The Lagrangian Grassmannian has a stratification by projected Richardson varieties, which are the type $C$ analogs of positroid varieties. We define type $C$ generalizations of bounded affine permutations and $k$-Bruhat intervals, as well as several other combinatorial posets which index positroid varieties. In addition, we generalize Postnikov's network parametrizations to projected Richardson varieties in $\Lambda(2n)$. In particular, we show that restricting the edge weights of our networks to $\mathbb{R}^+$ yields a family of parametrizations for totally nonnegative cells in $\Lambda(2n)$. In the process, we obtain a set of linear relations among the Pl\"ucker coordinates on $\text{Gr}(n,2n)$ which cut out the Lagrangian Grassmannian set-theoretically.

Abstract:
In this paper we study the action of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator in the Lagrangian Grassmannian manifold.

Abstract:
Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative structure is determined by the ring of (Q^~)-polynomials. We formulate a "quantum Schubert calculus" which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing the structure constants appearing in the quantum product of Schubert classes.

Abstract:
Given a complex structure $J$ on a real (finite or infinite dimensional) Hilbert space $H$, we study the geometry of the Lagrangian Grassmannian $\Lambda(H)$ of $H$, i.e. the set of closed linear subspaces $L\subset H$ such that $$J(L)=L^\perp.$$ The complex unitary group $U(H_J)$, consisting of the elements of the orthogonal group of $H$ which are complex linear for the given complex structure, acts transitively on $\Lambda(H)$ and induces a natural linear connection in $\Lambda(H)$. It is shown that any pair of Lagrangian subspaces can be joined by a geodesic of this connection. A Finsler metric can also be introduced, if one regards subspaces $L$ as projections $p_L$ (=the orthogonal projection onto $L$) or symmetries $\e_L=2p_L-I$, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in $\Lambda(H)$: a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. We extend these results to the classical Banach-Lie groups of Schatten.

Abstract:
The Drinfel'd Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the canonical embedding of the Lagrangian Grassmannian. We show that the defining ideal of any Schubert subvariety of the Drinfel'd Lagrangian Grassmannian is generated by polynomials which give a straightening law on an ordered set. Consequentially, any such subvariety is Cohen-Macaulay and Koszul. The Hilbert function is computed from the straightening law, leading to a new derivation of certain intersection numbers in the quantum cohomology ring of the Lagrangian Grassmannian.

Abstract:
We give a embedding of the Lagrangian Grassmannian LG(n) inside an ordinary Grassmannian that is well-behaved with respect to the Wronski map. As a consequence, we obtain an analogue of the Mukhin-Tarasov-Varchenko theorem for LG(n). The restriction of the Wronski map to LG(n) has degree equal to the number of shifted or unshifted tableaux of staircase shape. For special fibres one can define bijections, which, in turn, gives a bijection between these two classes of tableaux. The properties of these bijections lead a geometric proof of a branching rule for the cohomological map H*(Gr(n,2n)) x H*(LG(n)) -> H*(LG(n)), induced by the diagonal inclusion LG(n) -> LG(n) x Gr(n,2n). We also discuss applications to the orbit structure of jeu de taquin promotion on staircase tableaux.

Abstract:
We introduce the equation of n-dimensional totally geodesic submanifolds of a manifold E as a submanifold of the second order jet space of n-dimensional submanifolds of E. Next we study the geometry of n-Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n-dimensional totally geodesic submanifolds. We define the n-Grassmannian structure as the equivalence class of such connections, recovering for n=1 the case of theory of projectively equivalent connections. By introducing the equation of parametrized n-dimensional totally geodesic submanifolds as a submanifold of the second order jet space of the trivial bundle on the space of parameters, we discover a relation of covering between the `parametrized' equation and the `unparametrized' one. After having studied symmetries of these equations, we discuss the case in which the space of parameters is equal to R^n.