相关论文: Operads, Grothendieck topologies and Deformation t…
Given a nonsymmetric operad $\mathcal{O}$, we first construct two new nonsymmetric operads $\mathcal{O}^{\mathrm{comp}}$ and $\mathcal{O}^{\mathrm{Dend}}$. These operads are respectively useful to study compatible and split Loday-algebras.…
It is shown that every algebra over the chain operad of the little disks operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth manifold equipped with an integrable graded commutative associative product on the…
We explore how the higher order Hochschild cohomology controls a deformation theory when the simplicial set models the 3-sphere. Besides generalizing to the $d$-sphere for any $d\geq1$, we also investigate a deformation theory corresponding…
Dendriform coalgebras are the dual notion of dendriform algebras and are splitting of associative coalgebras. In this paper, we define a cohomology theory for dendriform coalgebras based on some combinatorial maps. We show that the…
We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Groebner…
Motivated by various developments in algebraic combinatorics and its applications, we investigate here the fine structure of a fundamental but little known theorem, the Gerstenhaber and Schack cohomology comparison theorem.The theorem…
To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW…
A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish…
In this note, we present a few existence theorems for the quotient of a scheme by the action of a group. The first two sections are devoted to Grothendieck topologies and descent theory. The third one is dealing with quotients: we first…
Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--Witten invariants, and the role of this story in…
We begin with a deformation of a differential graded algebra by adding time and using a homotopy. It is shown that the standard formulae of It\^o calculus are an example, with four caveats: First, it says nothing about probability. Second,…
The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical…
This paper studies the formal deformations of differential algebra morphisms. As a consequence, we develop a cohomology theory of differential algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Then,…
Inspired by the work of Wang and Zhou [4] for Rota-Baxter algebras, we develop a cohomology theory of Rota-Baxter systems and justify it by interpreting the lower degree cohomology groups as formal deformations and as abelian extensions of…
Given a finite $\mathbb{Z}_2$-graded group $\hat{\mathsf{G}}$ with ungraded subgroup $\mathsf{G}$ and a twisted cocycle $\hat{\lambda} \in Z^n(B \hat{\mathsf{G}}; \mathsf{U}(1)_{\pi})$ which restricts to $\lambda \in Z^n(B \mathsf{G};…
We use homotopy operators for the $L_\infty$-algebra associated with an equivariant deformation problem in order to describe a smooth parametrization of the space of structures around a given one. Along the way we give new algebraic and…
Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental…
The paper is devoted to a generalized and improved version of author's approach to Gromov bounded cohomology theory. In particular, the awkward countability assumption is removed and the aspects related to homological algebra are clarified.…
Recently, Grosse and Lechner introduced a novel deformation procedure for non-interacting quantum field theories, giving rise to interesting examples of wedge-localized quantum fields with a non-trivial scattering matrix. In the present…
Given a metrizable topological vector space, we can also use its von Neumann bornology or its bornology of precompact subsets to do analysis. We show that the bornological and topological approaches are equivalent for many problems. For…