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Galilean conformal algebras can be constructed by contracting a finite number of conformal algebras, and enjoy truncated $\mathbb{Z}$-graded structures. Here, we present a generalisation of the Galilean contraction procedure, giving rise to…
In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is…
Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and…
Atiyah and Hirzebruch gave examples ofeven degree torsion classes in the singularcohomology of a smooth complex projective manifold, which arenot Poincar\'{e} dual to an algebraiccycle. We notice that the order ofthese classes are small…
An Anosov flow on a smooth three-manifold $M$ gives rise to a Liouville structure on $\mathbb{R} \times M$ by a construction of Mitsumatsu. In a recent paper, Cieliebak, Lazarev, Massoni and Moreno ask whether an embedded torus $\Sigma…
Under certain assumptions (such as weak exacteness or monotonicity) we show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some strong forms of…
In the framework of Special Bohr - Sommerfeld geometry it was established that an ample divisor in compact algebraic variety can define almost canonically certain real submanifold which is lagrangian with respect to the corresponding Kahler…
The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in K\"ahler manifolds with real holomorphy potentials. Examples of submanifolds of this kind incuding soliton solutions for Lagrangian mean…
Several results in recent years have shown that the usual generalizations of taut foliations to higher dimensions, based only on topological concepts, lead to a theory that lacks the complexity of its 3-dimensional counterpart. Instead, we…
In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical…
We introduce a general technique to construct Lagrangian torus fibrations in degenerations of K\"ahler manifolds. We show that such torus fibrations naturally occur at the boundary of the A'Campo space. This space extends a degeneration…
A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential…
We review the construction of Lagrangians for higher spin fields of mixed symmetry in the framework of graded geometry. The main advantage of the graded formalism in this context is that it provides universal expressions, in the sense that…
We associate a natural $\lambda$-family ($\lambda \in \R \setminus \{0\} $) of flat Lagrangian immersions in $\C^n$ with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same…
We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a monoidal category, which is a version of the "category" of symplectic manifolds and canonical relations obtained by localizing them…
Consider a Lagrangian fibration $\pi\colon X\to \mathbb P^n$ on a hyperk\"ahler manifold $X$. There are two ways to construct a holomorphic family of deformations of $\pi$ over $\mathbb C$. The first one is known under the name…
We study the Lagrangian isotopy classification of Lagrangian spheres in the Milnor fibre, $B_{d,p,q}$, of the cyclic quotient surface T-singularity $\frac{1}{dp^2} (1,dpq-1)$. We prove that there is a finitely generated group of…
We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of \cite{CO}. Related $\AI$-formulas hold for transversal choice of chains. Two different computations are…
This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally…
Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a…