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Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($C^{2n}) are investigated under suitable continuity restrictions on cochains. The first and second cohomology…

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces $\mathbb{A}^{m,n}$ with trivial coefficients. Previously known results covered the classical case $\mathbb{A}^m = \mathbb{A}^{m,0}$ as well as the case…

For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we…

We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the…

The analogy between 1+3 splittings of the spacetime tangent bundle and the splitting of the tangent bundle to the bundle of linear frames into vertical and horizontal sub-bundles is described from the unifying standpoint of the geometry of…

We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed…

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of…

We prove that for a system of indeterminates (X_a) indiced by the P^2(2), the projective plane over F_2, there exists a 3-3 correspondance compatible with the incidence structures of P^2(2), such that (X_a) is one of the orbits of it. We…

Into this note we collect topics related to homogeneous vector bundles, elliptic adjoint orbits and so forth.

For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of $\mathbb{R}^n$. We also prove a formula which computes…

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a…

We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the…

We construct and identify star representations canonically associated with holonomy reducible simple symplectic symmetric spaces. This leads the a non-commutative geometric realization of the correspondence between causal symmetric spaces…

We collect evidence that the notion of path-ordered non-abelian integration admits an extension to two dimensions. We propose the corresponding notion of non-abelian 2-form along the lines of Lie algebroid theory and argue it is an…

This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the \'etale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois…

Let $L/K$ be a finite Galois extension of complete discrete valued fields of characteristic $p$. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer $n\geq 0$, let $W_n(\sO_L)$ denote the ring of Witt…

Let $k$ be a field, $C\to \Spec k$ be a stable curve and let $G$ be a finite group acting faithfully on the curve $C\to \Spec k$. In this article, we compute the vector space $\Ext^1_G(\Omega_{C/k}, \O_C)$, the sheaf $\Omega_{C/k}$ being…

The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to…

We define a bicategory with \'etale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual…

The main purpose of this paper is to provide a full cohomology of a Hom-pre-Lie algebra with coefficients in a given representation. This new type of cohomology exploit strongly the Hom-type structure and fits perfectly with simultaneous…