Related papers: Equivariant Quotients of Derived Symplectic Spaces…
We construct the derived Legendrian category $\mathcal{F}_{c}(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty,2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with…
Motivated by the cohomological construction for the BV formalism from physics, this thesis asks how to perform the intersections and quotients appearing in the BV construction. This leads to the study of the derived symplectic reduction and…
This is a survey paper on derived symplectic geometry, that will appear as a chapter contribution to the book "New Spaces for Mathematics and Physics", edited by Mathieu Anel and Gabriel Catren. Our goal is to explain how derived stacks can…
We study when the derived intersection of two smooth subvarieties of a smooth variety is formal. As a consequence we obtain a derived base change theorem for non-transversal intersections. We also obtain applications to the study of the…
We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a…
This is the third installment in a series of papers on the subject of derived contact structures. In this paper, we formally introduce the notion of a Legendrian structure in the derived context and provide natural constructions. We then…
Sivek proves a "van Kampen" decomposition theorem for the combinatorial Legendrian contact algebra (also known as the Chekanov-Eliashberg algebra) of knots in standard contact $\R^3$ . We prove an analogous result for the holomorphic curve…
Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat…
We will quickly explore the derived geometry of zero loci of sections of vector bundles, with particular emphasis on derived critical loci. In particular we will single out many of the derived geometric structures carried by derived…
We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction…
The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived…
We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic…
Derived geometry provides powerful tools to handle non-transverse intersections and singular moduli problems arising in geometry and theoretical physics. While derived algebraic geometry has been extensively developed, classical field…
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…
Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb C$. Consider the Liouville symplectic structure on the total space $T^*G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal…
In this paper we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two…
In this paper, we use derived sheaves to study rigidity phenomena in the cotangent bundles of manifolds endowed with some locally conformally symplectic ($\frak{lcs}$) structure. Taking inspiration from the work of Guillermou, Kashiwara and…
We introduce and study the derived moduli stack $\mathrm{Symp}(X,n)$ of $n$-shifted symplectic structures on a given derived stack $X$, as introduced by [PTVV] (IHES Vol. 117, 2013). In particular, under reasonable assumptions on $X$, we…
Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is…
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks.…