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A system with anholonomic constraints where the trajectories of physical degrees of freedom are autoparallels on a manifold equipped with a general Cartan connection is discussed. A variational principle for the autoparallel trajectories is…
We prove a version of quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal D$-modules on the stack of rank $N$…
Deligne constructed a remarkable local system on $\bP^1-\{0,\infty\}$ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic…
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of…
We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local…
This note aims to clarify the deep relationship between birational modifications of a variety and semiorthogonal decompositions of its derived category of coherent sheaves. The result is a conjecture on the existence and properties of…
If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard…
This paper introduces the notion of locally algebraic representations and corresponding sheaves in the context of the cohomology of arithmetic groups. These representations are of relevance for the study of integral structures and special…
The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the…
We extend the usual internal logic of a (pre)topos to a more general interpretation, called the stack semantics, which allows for "unbounded" quantifiers ranging over the class of objects of the topos. Using well-founded relations inside…
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…
We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.
We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived infinity-category…
For a variety over certain topological rings $R$, like $\mathbb{Z}_p$ or $\mathbb{C}$, there is a well-studied way to topologize the $R$-points on the variety. In this paper, we generalize this definition to algebraic stacks. For an…
In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra, A, a topos T(A) of sheaves on a locale and a commutative C*-algebra, a, within that topos. The Gelfand spectrum…
In this paper we explore the link between the theory of sheaves on graphs and noncommutative geometry showing that many concepts and constructions in the latter can be generalized and enhanced using methods coming from the former. They…
We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new…
We study the bounded derived categories of torus-equivariant coherent sheaves on smooth toric varieties and Deligne-Mumford stacks. We construct and describe full exceptional collections in these categories. We also observe that these…
We formulate a connection between a topological and a geometric category. The former is the idempotent completion of the (horizontal) trace of the affine Hecke category, while the latter is the equivariant derived category of the…
This paper develops the tools of formal algebraic geometry in the setting of noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a…