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We provide a framework which generalizes algebraic models of a homotopy theory of spaces to the genuine equivariant case for a discrete group. We explain how this applies to commutative differential graded algebra (cdga) models and complete…

We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo…

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean…

Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since…

We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as…

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…

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph…

We study the equation E_fc of flat connections in a fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an…

We study the space of pseudo-holomorphic spheres in compact symplectic manifolds with convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This…

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on…

For a finite group $D$, we study categorical factorisation homology on oriented surfaces equipped with principal $D$-bundles, which `integrates' a (linear) balanced braided category $\mathcal{A}$ with $D$-action over those surfaces. For…

We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural…

It is well known that the moduli space of flat connections on a trivial principal bundle MxG, where G is a connected Lie group, is isomorphic to the representation variety Hom(\pi_1(M), G)/G. For a tiling T, viewed as a marked copy of R^d,…

We construct stable vector bundles on the space of symmetric forms of degree d in n+1 variables which are equivariant for the action of SL_{n+1}(C), and admit an equivariant free resolution of length 2. For n=1, we obtain new examples of…

For Hamiltonian circle actions on compact, connected, four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This…

We generalize the higher Riemann-Hilbert correspondence in the presence of scalar curvature for a (possibly non-compact) smooth manifold $M$. We show that the dg-category of curved $\infty$-local systems, the dg-category of graded vector…

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological…

In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we…

Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply-connected space $X$ there is a dg Lie algebra $\Lambda_X$ and a (coassociative…

Given a fiber bundle, we construct a differential graded Lie algebra model for the classifying space of the monoid of homotopy equivalences of the base covered by a fiberwise isomorphism of the total space.