20 research outputs found
An Overview of Topological and Fuzzy Topological Hypergroupoids
On a hypergroup, one can define a topology such that the hyperoperation is pseudocontinuous or continuous.This concepts can be extend to the fuzzy case and a connection between the classical and the fuzzy (pseudo)continuous hyperoperations can be given.This paper, that is his an overview of results received by S. Hoskova-Mayerova with coauthors I. Cristea , M. Tahere and B. Davaz, gives examples of topological hypergroupoids and show that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. In particular, it shows a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation
Augmented Homotopical Algebraic Geometry
We develop the framework for augmented homotopical algebraic geometry. This
is an extension of homotopical algebraic geometry, which itself is a
homotopification of classical algebraic geometry. To do so, we define the
notion of augmentation categories, which are a special class of generalised
Reedy categories. For an augmentation category, we prove the existence of a
closed Quillen model structure on the presheaf category which is compatible
with the Kan-Quillen model structure on simplicial sets. Moreover, we use the
concept of augmented hypercovers to define a local model structure on the
category of augmented presheaves. We prove that crossed simplicial groups, and
the planar rooted tree category are examples of augmentation categories.
Finally, we introduce a method for generating new examples from old via a
categorical pushout construction.Comment: 36 pages, comments welcom
An introduction to derived (algebraic) geometry
These are notes from an introductory lecture course on derived geometry,
given by the second author, mostly aimed at an audience with backgrounds in
geometry and homological algebra. The focus is on derived algebraic geometry,
mainly in characteristic , but we also see the tweaks which extend most of
the content to analytic and differential settings. The main motivating
applications given are in moduli theory, with practically applicable
representability theorems.Comment: 93pp; v2 minor changes; v3 minor additions, mostly reference
Presenting higher stacks as simplicial schemes
We show that an n-geometric stack may be regarded as a special kind of
simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where
surjectivity is defined in terms of covering maps, yielding Artin n-stacks,
Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This
formulation adapts to all HAG contexts, so in particular works for derived
n-stacks (replacing rings with simplicial rings). We exploit this to describe
quasi-coherent sheaves and complexes on these stacks, and to draw comparisons
with Kontsevich's dg-schemes. As an application, we show how the cotangent
complex controls infinitesimal deformations of higher and derived stacks.Comment: 55 pages; v3 content rearranged with many corrections; final version,
to appear in Adv. Math; v4 corrections in section 7.
Presenting higher stacks as simplicial schemes
We show that an n-geometric stack may be regarded as a special kind of
simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where
surjectivity is defined in terms of covering maps, yielding Artin n-stacks,
Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This
formulation adapts to all HAG contexts, so in particular works for derived
n-stacks (replacing rings with simplicial rings). We exploit this to describe
quasi-coherent sheaves and complexes on these stacks, and to draw comparisons
with Kontsevich's dg-schemes. As an application, we show how the cotangent
complex controls infinitesimal deformations of higher and derived stacks.Comment: 55 pages; v3 content rearranged with many corrections; final version,
to appear in Adv. Math; v4 corrections in section 7.
Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras
We consider the Hecke pair consisting of the group of affine
transformations of a number field that preserve the orientation in every
real embedding and the subgroup consisting of transformations with
algebraic integer coefficients. The associated Hecke algebra
has a natural time evolution , and we describe the corresponding phase
transition for KMS-states and for ground states. From work of
Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated
to has an essentially unique arithmetic subalgebra. When we import this
subalgebra through the isomorphism of to a corner in the
Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an
arithmetic subalgebra of on which ground states exhibit the
`fabulous' property with respect to an action of the Galois group
, where is the narrow Hilbert class field.
In order to characterize the ground states of the -dynamical system
, we obtain first a characterization of the ground
states of a groupoid -algebra, refining earlier work of Renault. This is
independent from number theoretic considerations, and may be of interest by
itself in other situations.Comment: 21 pages; v2: minor changes and correction
Extended graph of the fuzzy topographic topological mapping model
Fuzzy topological topographic mapping (TTTM) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, TTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of TTTM, namely the homeomorphisms between its components, allows the generation of new TTTM. The generated FTTMs can be represented as pseudo graphs. A graph of pseudo degree zero is a special type of graph where each of the FTTM components differs from the one adjacent to it. Previous researchers have investigated and conjectured the number of generated FTTM pseudo degree zero with respect to n number of components and k number of versions. In this paper, the conjecture is proven analytically for the first time using a newly developed grid-based method. Some definitions and properties of the novel grid‐based method are introduced and developed along the way. The developed definitions and properties of the method are then assem-bled to prove the conjecture. The grid‐based technique is simple yet offers some visualization fea-tures of the conjecture
Derived moduli of complexes and derived Grassmannians
In the first part of this paper we construct a model structure for the
category of filtered cochain complexes of modules over some commutative ring
and explain how the classical Rees construction relates this to the usual
projective model structure over cochain complexes. The second part of the paper
is devoted to the study of derived moduli of sheaves: we give a new proof of
the representability of the derived stack of perfect complexes over a proper
scheme and then use the new model structure for filtered complexes to tackle
moduli of filtered derived modules. As an application, we construct derived
versions of Grassmannians and flag varieties.Comment: 54 pages, Section 2.4 significantly extended, minor corrections to
the rest of the pape