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We produce twisted derived equivalences between torsors under abelian varieties and their moduli spaces of simple semi-homogeneous sheaves. We also establish the natural converse to this result and show that a large class of twisted derived…
The filter quotient construction is a particular instance of a filtered colimit of categories. It has primarily been considered in the context of categorical logic, where it has been used effectively to construct non-trivial models, for…
We study Le Potier's strange duality conjecture for moduli spaces of sheaves over generic abelian surfaces. We prove the isomorphism for abelian surfaces which are products of elliptic curves, when the moduli spaces consist of sheaves of…
We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This…
We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of decomposable maximal Cohen-Macaulay modules of higher rank. A result of Klyachko…
We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus, and Schauder's fixed…
We have recently showed that it is possible to deal with collections of indistinguishable elementary particles (in the context of quantum mechanics) in a set-theoretical framework by using hidden variables, in a sense. In the present paper…
We study classes of dynamical systems that can be obtained by constructing recursive networks with monotone Boolean functions. Stack filters in nonlinear signal processing are special cases of such systems. We show an analytical connection…
The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show…
We partially resolve conjectures of Deligne and Simpson concerning $\mathbb{Z}$-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with $\mathbb{Q}$-anisotropic monodromy,…
We identify a class of "quasi-compact semi-separated" (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic…
We introduce and develop the theory of metric sheaves. A metric sheaf $\A$ is defined on a topological space $X$ such that each fiber is a metric model. We describe the construction of the generic model as the quotient space of the sheaf…
We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ contains at least two non-quasi-isometric…
We introduce the notion of a "graded topological space": a topological space endowed with a sheaf of abelian groups which we think of as a sheaf of gradings. Any object living on a graded topological space will be graded by this sheaf of…
In view of applications to the construction of moduli spaces of objects in algebraic supergeometry, we start a systematic study of stacks in that context. After defining a superstack as a stack over the \'etale site of superschemes, we…
The local analytic classification of irregular linear q-difference equations (Ramis-Sauloy-Zhang) involves the classfication of filtered q-difference modules with a prescribed associated graded module. We prove in a more general setting the…
The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper.…
We study the categories of discrete modules for topological rings arising as the rings of operations in various kinds of topological K-theory. We prove that for these rings the discrete modules coincide with those modules which are locally…
We study the moduli of G-local systems on smooth but not necessarily proper complex algebraic varieties. We show that, when suitably considered as derived algebraic stacks, they carry natural Poisson structures, generalizing the well known…
We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.