Related papers: A configuration space model for algebraic function…
Using log-geometry, we construct a model for the configuration category of a smooth algebraic variety. As an application, we prove the formality of certain configuration spaces.
We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known…
We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples,…
We prove the validity over $\mathbb{R}$ of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts--Stanley. We get as a result that the real…
We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to…
It is a well established fact, that any projective algebraic variety is a moduli space of representations over some finite dimensional algebra. This algebra can be chosen in several ways. The counterpart in algebraic geometry is…
In this article, we construct operator models for meromorphic functions of bounded type on Krein spaces. This construction is based on certain reproducing kernel Hilbert spaces which are closely related to model spaces. Specifically, we…
After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define…
We construct a real combinatorial model for the configuration spaces of points of compact smooth oriented manifolds without boundary. We use these models to show that the real homotopy type of configuration spaces of a simply connected such…
In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed…
We compute small rational models for configuration spaces of points on oriented surfaces, as right modules over the framed little disks operad. We do this by splitting these surfaces in unions of several handles. We first describe rational…
A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow…
We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set $Z\subset \mathbb{A}^{n-2}\subset \mathbb{A}^{n}$, we construct an…
We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary, we obtain the algebraicity of the stack of…
We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack.…
It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real…
We show that in codimension at least 3, spaces of locally flat topological embeddings of manifolds are correctly modelled by derived spaces of maps between their configuration categories (under mild smoothability conditions). That general…
We consider A-hypergeometric functions associated to normal sets in the plane. We give a classification of all point configurations for which there exists a parameter vector such that the associated hypergeometric function is algebraic. In…
We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of…
We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…