Related papers: A Representability Theorem for Stacks in Derived G…
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks.…
We study Hom 2-functors parameterizing 1-morphisms of algebraic stacks, and prove that it is representable by an algebraic stack under certain conditions, using Artin's criterion. As an application we study Picard 2-functors which…
For algebraic stacks over number fields, we define their Brauer-Manin sets, Brauer-Manin pairings, and extend the descent theory of Colliot-Th\'el\`ene and Sansuc. By extending Sansuc's exact sequence, we show the torsionness of Brauer…
The purpose of this paper is to develop a theory of $(\infty, 1)$-stacks, in the sense of Hirschowitz-Simpson's `Descent Pour Les n-Champs', using the language of quasi-category theory and the author's local Joyal model structure. The main…
G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles. Analogous obstruction…
There is growing evidence that independently trained AI systems come to represent the world in the same way. In other words, independently trained embeddings from text, vision, audio, and neural signals share an underlying geometry. We call…
Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes ${Rep_n(A)}$ parametrizing the finite-dimensional represen- tations of a noncommutative algebra A should be…
We establish a unified group-theoretic framework bridging the arithmetic homotopy exact sequence of a variety and the Birman exact sequence of a surface. Within this framework, we reinterpret classical arithmetic notions - such as the…
We develop a characterisation of non-Archimedean derived analytic geometry based on dg enhancements of dagger algebras. This allows us to formulate derived analytic moduli functors for many types of pro-\'etale sheaves, and to construct…
For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a…
We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of…
We use categorical method and birational geometry to study moduli spaces of quiver representations. From certain "representable" functor, we construct a birational transformation from the moduli space of representations of one quiver to…
The aim of this paper is to study the structure of the higher-dimensional Teichm\"uller and Riemann moduli spaces, viewed as stacks over the category of complex manifolds. We first show that the space of complex operators on a smooth…
We give a geometric formulation of To\"en's derived Hall algebra by constructing Grothendieck's six operations for the derived category of lisse-\'etale constructible sheaves on the derived stacks of complexes. Our formulation is based on…
We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a $\Theta$-stratification. These results provide a…
We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud's relative representability theorem, we give sufficient conditions…
We define a category whose objects are finite etale coverings of an algebraic stack and prove that it is a Galois category and that it allows one to compute the fundamental group of the stack. We then prove a Van Kampen theorem for…
The Geometrical Lemma is a classical result in the theory of (complex) smooth representations of $p$-adic reductive groups, which helps to analyze the parabolic restriction of a parabolically induced representation by providing a filtration…
We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the…
We present a method for compactifying stacks of $\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are…