81 research outputs found
Countable Short Recursively Saturated Models of Arithmetic
Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated
The homotopy theory of coalgebras over a comonad
Let K be a comonad on a model category M. We provide conditions under which
the associated category of K-coalgebras admits a model category structure such
that the forgetful functor to M creates both cofibrations and weak
equivalences.
We provide concrete examples that satisfy our conditions and are relevant in
descent theory and in the theory of Hopf-Galois extensions. These examples are
specific instances of the following categories of comodules over a coring. For
any semihereditary commutative ring R, let A be a dg R-algebra that is
homologically simply connected. Let V be an A-coring that is semifree as a left
A-module on a degreewise R-free, homologically simply connected graded module
of finite type. We show that there is a model category structure on the
category of right A-modules satisfying the conditions of our existence theorem
with respect to the comonad given by tensoring over A with V and conclude that
the category of V-comodules in the category of right A-modules admits a model
category structure of the desired type. Finally, under extra conditions on R,
A, and V, we describe fibrant replacements in this category of comodules in
terms of a generalized cobar construction.Comment: 34 pages, minor corrections. To appear in the Proceedings of the
London Mathematical Societ
Finding groups in Zariski-like structures
We study quasiminimal classes, i.e. abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. For these classes, we develop an independence notion, and in particular, a theory of independence in M^{eq}. We then generalize Hrushovski's Group Configuration Theorem to our setting. In an attempt to generalize Zariski geometries to the context of quasiminimal classes, we give the axiomatization for Zariski-like structures, and as an application of our group configuration theorem, show that groups can be found in them assuming that the pregeometry obtained from the bounded closure operator is non-trivial. Finally, we study the cover of the multiplicative group of an algebraically closed field and show that it provides an example of a Zariski-like structure
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Relative p-adic Hodge theory: Foundations
We describe a new approach to relative p-adic Hodge theory based on
systematic use of Witt vector constructions and nonarchimedean analytic
geometry in the style of both Berkovich and Huber. We give a thorough
development of phi-modules over a relative Robba ring associated to a perfect
Banach ring of characteristic p, including the relationship between these
objects and etale Z_p-local systems and Q_p-local systems on the algebraic and
analytic spaces associated to the base ring, and the relationship between etale
cohomology and phi-cohomology. We also make a critical link to mixed
characteristic by exhibiting an equivalence of tensor categories between the
finite etale algebras over an arbitrary perfect Banach algebra over a
nontrivially normed complete field of characteristic p and the finite etale
algebras over a corresponding Banach Q_p-algebra. This recovers the
homeomorphism between the absolute Galois groups of F_p((pi)) and
Q_p(mu_{p^infty}) given by the field of norms construction of Fontaine and
Wintenberger, as well as generalizations considered by Andreatta, Brinon,
Faltings, Gabber, Ramero, Scholl, and most recently Scholze. Using Huber's
formalism of adic spaces and Scholze's formalism of perfectoid spaces, we
globalize the constructions to give several descriptions of the etale local
systems on analytic spaces over p-adic fields. One of these descriptions uses a
relative version of the Fargues-Fontaine curve.Comment: 210 pages; v5: version to appear in Asterisqu
Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations
New striking analogies between H. Hahn’s fields of generalised series with real coefficients, G. H. Hardy’s field of germs of real valued functions, and J. H. Conway’s field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery
C*-Algebren
The theory of C*-algebras plays a major role in many areas of modern mathematics, like Non-commutative Geometry, Dynamical Systems, Harmonic Analysis, and Topology, to name a few. The aim of the conference “C*-algebras” is to bring together experts from all those areas to provide a present day picture and to initiate new cooperations in this fast growing mathematical field
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