Related papers: $A_\infty$-Minimal Model on Differential Graded Al…

The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater…

De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod $v$ currents are smooth outside of a singular set of codimension at least $1.$…

The Schouten tensor \ $A$ \ of a Riemannian manifold \ $(M,g)$ provides important scalar curvature invariants $\sigma_k$, that are the symmetric functions on the eigenvalues of $A$, where, in particular, $\sigma_1$ \ coincides with the…

We give necessary conditions for the existence of a compact manifold locally modelled on a given homogeneous space, which generalize some earlier results, in terms of relative Lie algebra cohomology. Applications include both reductive and…

Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(T)/T|$ is invertible in the field $k$ then the algebra of cochains $C^*(BG;k)$ is formal as an $A_\infty$ algebra, or equivalently as a DG algebra.

Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The…

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections…

We settle several questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. An important new technical ingredient introduced in this paper is an…

We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling…

Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.

A compact 4-dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing T^2-bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the…

A differential algebra of finite type over a field k is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated k-algebra. The prototypical example is the algebra of…

This thesis is a study of various ways of measuring the size and complexity of finitely generated modules over a Noetherian local ring. The classical example is the multiplicity or degree. Here we investigate several variants of the degree…

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$. Let $\mathcal{D}^2_{fg}(A)$ denote the derived category of $2$-periodic complexes with finitely generated cohomology modules. Let $\mathcal{K}^2(\proj A) $ denote the…

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum of the inverse of the canonical bundle…

The paper presents a classification theorem for the class of flat connections with triangular (0,1)-components on a topologically trivial complex vector bundle over a compact Kahler manifold. As a consequence we obtain several results on…

Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^3=0\ne\mathfrak{m}^2$. Set $k=R/\mathfrak{m}$ and $e=\text{rank}_{k}(\mathfrak{m}/\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated…

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These…

We show that when $A$ is a reduced algebra over a characteristic zero field $k$ and the module of K\"ahler differentials $\Omega_{A/k}=0$, then $A$ is ind-\'etale, partially answering a question of Bhatt. As further applications of this…

The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The task is accomplished in three steps. The first step is the construction of a modified cobar complex adapted to a…