Related papers: Presenting higher stacks as simplicial schemes

We consider the stratification of the category of smooth representations of $\mbox{GL}_n(F)$ (for $F$ a $p$-adic field) induced by degenerate Whittaker models. We show that, remarkably, the successive quotient categories in this…

Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…

We propose an abelian categorification of $\hat{Z}$-invariants for Seifert $3$-manifolds. First, we give a recursive combinatorial derivation of these $\hat{Z}$-invariants using graphs with certain hypercubic structures. Next, we consider…

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…

In this article we give a new proof of Ng\^o's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the…

Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a…

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold diagrams, which are stratifications of…

This paper develops a unified framework for observables in n-plectic geometry, extending the L_infty-algebra of Hamiltonian (n-1)-forms to Hamiltonian forms of all degrees via a degree-shifting Grassmann variable u that encodes submanifold…

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains.…

We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey…

Fix a scheme $X$ over a field of characteristic zero that is equipped with an action of a reductive algebraic group $G$. We give necessary and sufficient conditions for a $G$-equivariant coherent sheaf on $X$ or a bounded-above complex of…

In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely specify the stack. Our main result shows that…

The purpose of this thesis is to introduce two new kinds of hyperplane arrangements, inspired by the graphic arrangements and $q$-deformations of graphic arrangements. In this thesis, the author extends the definition of $q$-deformation to…

Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$…

We prove that shifted cotangent stacks carry a canonical shifted symplectic structure. We also prove that shifted conormal stacks carry a canonical Lagrangian structure. These results were believed to be true but no written proof was…

In this note I introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group $G$ as a sheaf on some geometric object (a stack). This point of…

Let $\mathcal{X}$ be a smooth Deligne-Mumford stack which is generically a scheme and has quasi-projective coarse moduli. If $\mathcal{X}$ has elementary Abelian 2-group stabilizers and the coarse moduli of the inertia stack is smooth, we…

Let $A$ be a commutative ring, let $k\in\mathbb{Z}^+$, and let $\vec{s}=(n_1,\dots,n_k)\in(\mathbb{Z}^+)^k$ with $n=\min_a(n_a)-1$. We attach to $\vec{s}$ a diagonal simplicial tensor module $X_\bullet(\vec{s};A)$ whose $p$-simplices are…

We propose an algebro-geometric interpretation of the Schur and Macdonald indices of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs). We conjecture that there exists an affine scheme $X$, which reduces to the Higgs…

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the…