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By a theorem of Bernhard Keller the de Rham cohomology of a smooth variety is isomorphic to the periodic cyclic homology of the differential graded category of perfect complexes on the variety. Both the de Rham cohomology and the cyclic…
This work focuses on approximation and generation for the derived category of complexes with quasi-coherent cohomology on algebraic stacks. Our methods establish that approximation by compact objects descends along covers that are…
In the field of solving partial differential equations (PDEs), Hilbert complexes have become highly significant. Recent advances focus on creating new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as shown by Arnold and Hu…
We introduce a Hilbert $A$-module structure on the higher oscillatory module, where $A$ denotes the $C^*$-algebra of bounded endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
A regular realizability (RR) problem is testing nonemptiness of intersection of some fixed language (filter) with given regular language. We study here complexity of RR problems. It appears that for any language L there exists RR problem…
We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear…
In this second part of our series of papers, we develop an abstract framework suitable for de Rham complexes that depend on a parameter belonging to an arbitrary Banach space. Our primary focus is on spectral perturbation problems and the…
The goal of this paper is to derive a simple recursion that generates a sequence of fractions approximating $\sqrt[n]{k}$ with increasing accuracy. The recursion is defined in terms of a series of first-order non-linear difference equations…
In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. Recently, a series of paper has been published for analysis of expansion complexity and for testing sequences in terms of this new…
It is an important fact that extremal discrete structures -- that is, discrete structures of maximal size among those that avoid certain configurations -- exhibit strong pseudorandom behavior. We present instances of this phenomenon in the…
This paper investigates the relationship between the solvability of first-order differential equations and the topology of the underlying domain through the lens of de\,Rham cohomology. We analyze the conditions under which a closed 1-form…
The idea of a space with smooth structure is a generalization of an idea of a manifold. K. T. Chen introduced such a space as a differentiable space in his study of a loop space to employ the idea of iterated path integrals…
In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open…
We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise…
This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the "topological recursion", a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall…
We study differential $p$-forms on non-smooth and possibly fractal metric measure spaces, endowed with a local Dirichlet form. Using this local Dirichlet form, we prove a result on the localization of antisymmetric functions of $p+1$…
Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto,…
We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward…
This paper discusses the construction of local bounded commuting projections for discrete subcomplexes of the gradgrad complexes in two and three dimensions, which play an important role in the finite element theory of elasticity (2D) and…