arXiv research

New contact structures defined on differentiable stacks.

Introduces derived Lie n-groupoids with shifted symplectic structures.

New models for symplectic structures on classifying stacks.

New concept of coisotropic structures for differentiable stacks defined.

Proof that m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.

This thesis extends contact structures to differentiable stacks using line bundle-valued 1-forms.

The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.

Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetri…

The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.

A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed $2$-forms. It is shown that any derived scheme over $\mathbb{C}$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally …

This work explores symplectic structures on graded manifolds and higher Lie groupoids.

This paper puts the theory of quasi-Hamiltonian reduction in the framework of shifted symplectic structures developed by Pantev, Toën, Vaquié and Vezzosi. We compute the symplectic structures on mapping stacks and show how the AKSZ topological field theory defined by Calaque allows one to neatly package the constructio…

Continues work on derived manifolds and symplectic schemes, constructing virtual classes.

Defines duals of higher vector bundles for Lie 2-groupoids.

We prove a version of the affine Kempf-Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkahler quotients of T*G, where G is a complex reductive group.

The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids …

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to …

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schroding…

A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented b…

Let $({\bf X},ω_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that …

A new metric is created on a special bundle.

We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop…

Introduces a new method for symplectic reduction along submanifolds.

We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #m CP^2 # n bar(CP^2)…

Log-symplectic structures are Poisson structures $π$ on $X^{2n}$ for which $\bigwedge^n π$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…

Constructs cohomology decompositions for symmetric stacks.

Symplectic structures simplified for compact manifolds.

Symplectic 4-manifolds can be divided into three parts with a special structure.

The paper explores new algebraic structures and morphisms in graded settings.

This work evaluates graph models' robustness to structural distributional shifts.

Study on symplectic structures and their deformations.

Paper proves h-principles for symplectic structures and foliations.

The paper explores left invariant k-symplectic structures on Lie groups with bi-invariant metrics.

Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.

The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show t…

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

Found a new connected component in symplectic structures.

Symplectic structure found on projective structures on surfaces with boundary.

Study symplectic structures on low dimensional 2-step nilmanifolds.

The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.

The paper defines conditions for a Riemannian structure on a symplectic quotient.

The paper classifies symplectic fillings of lens spaces and constructs cobordisms.

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

New method classifies symplectic structures on Lie groups.

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced on the base manifold.