Related papers: A Topological Reconstruction Theorem for $D^{\inft…

We construct a new model structure on the category of dg presheaves over a topological space $X$, obtained through the right Bousfield localization of the local projective model structure. The motivation for this construction arises from…

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated…

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated…

We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we generalize Leray-Hirsch, K\"{u}nneth and Poincar\'{e}-Serre duality theorems on them.…

We generalise Hinich's Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha: {\mathcal F} \to {\mathcal…

We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr's holomorphic almost periodic functions on tube domains or algebras of all…

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For…

We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric…

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate…

The so called theory of derived D-modules is an extension of classical D-modules to derived algebraic geometry, which uses the derived information of the base scheme. We prove that the three different definitions of derived D-modules, given…

We introduce a pro-\'etale geometric object $D_\infty$ arising naturally from the tower of Artin-Schreier extensions in characteristic 2, equipped with a canonical endofunctor $O$ whose fixed points correspond to automorphic representations…

We introduce a notion of formally \'etale $\mathbb{E}_{\infty}$-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a perfect…

Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz--Reidemeister…

Let $X$ be a smooth $p$-adic Stein space with free tangent sheaf. We use the notion of Hochschild cohomology for sheaves of Ind-Banach algebras developed in our previous work to study the Hochschild cohomology of the algebra of infinite…

We give a new construction of $p$-adic overconvergent Hilbert modular forms by using Scholze's perfectoid Shimura varieties at infinite level and the Hodge--Tate period map. The definition is analytic, closely resembling that of complex…

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce…

In order to look for a well-behaved counterpart to Dolbeault cohomology in D-complex geometry, we study the de Rham cohomology of an almost D-complex manifold and its subgroups made up of the classes admitting invariant, respectively…

In this PhD thesis, we give a new geometric approach to higher Teichm\"uller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The…

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the…

We describe the strong dual space $({\mathcal O} (D))^*$ for the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables over a bounded Lipschitz domain $D$ with connected boundary $\partial D$ (as usual, ${\mathcal…