On perturbations of Hilbert spaces and probability algebras with a generic automorphism

International audienceWe prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $\aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, the theory of atomless probability algebras equipped with a generic automorphism is $\aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable

The classification problem for automorphisms of C*-algebras

We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.Comment: 21 page

Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.Malliteoria tutkii matemaattisia struktuureita ja niiden kokoelmia. Malli eli struktuuri koostuu perusjoukosta ja tämän alkioiden välisiä suhteita kuvaavista funktioista ja relaatioista. Esimerkiksi kokonaisluvut yhteen- ja kertolaskufunktioineen muodostavat mallin. Klassinen malliteoria tutkii elementaariluokkia, eli malliluokkia, jotka voidaan kuvata ensimmäisen kertaluvun teorialla. Esimerkkejä tällaisista on runsaasti geometriassa ja algebrassa. Monet mielenkiintoiset malliluokat ovat kuitenkin epäelementaarisia, minkä takia malliteorian tutkimuskenttää on laajennettu. 1980-luvulla Shelah kehitti abstraktit elementaariluokat yleiseksi pohjaksi epäelementaaristen luokkien tutkimiselle. Niissä malleja ei tutkita formaalilla kielellä, vaan mallien välisille suhteille annetaan joukko ehtoja. Toinen malliteorian yleistämisen suunta on metristen struktuurien tutkiminen. Näiden tarkasteluun elementaarilogiikka ei sovellu, mutta niiden tutkimiseen on kehitetty muita lähestymistapoja. Viime aikoina on laajimmim käytetty ns. jatkuvaa logiikkaa, jossa kaksiarvologiikan asemesta totuusarvot sijoittuvat reaalilukuvälille [0,1]. Väitöskirjassa kehitetään uusi, Shelah'n abstrakteihin elementaariluokkiin pohjautuva lähestymistapa metriseen malliteoriaan. Abstraktin lähestymistavan suurin etu on mahdollisuus lisätä tutkittavaan struktuuriluokkaan kokoelma yleistettyjä isomorfismeja. Malliteoriassa isomorfismi on kuvaus, joka säilyttää mallin rakenteen tarkasti. Funktionaalianalyysissa isomorfismien sallitaan kuitenkin venyttää vektoreiden pituuksia hieman. Myös operaattoreiden approksimointia tarkasteltaessa nousee tarve yleisemmälle isomorfismin käsitteelle. Aiemmissa lähestymistavoissa yleistettyjä isomorfismeja on jouduttu tarkastelemaan malleihin jälkikäteen tehtävinä muutoksina. Uusi, sisäänrakennettu lähestymistapa mahdollistaa malliteoriassa tärkeiden työkalujen kehittämisen yleistettyjen isomorfismien suhteen. Väitöskirja keskittyy näistä riippumattomuuskäsitteen ja isolaation kehittämiseen. Yksi väitöskirjan päätuloksista osoittaa, että esitetty määritelmä antaa hyvinkäyttäytyvän riippumattomuuskäsitteen jo perturbaation suhteen omega-stabiileille malliluokille

On perturbations of continuous structures

We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $\aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations (where the notion of perturbation is part of the data). As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations

Spacetime-Free Approach to Quantum Theory and Effective Spacetime Structure

Motivated by hints of the effective emergent nature of spacetime structure, we formulate a spacetime-free algebraic framework for quantum theory, in which no a priori background geometric structure is required. Such a framework is necessary in order to study the emergence of effective spacetime structure in a consistent manner, without assuming a background geometry from the outset. Instead, the background geometry is conjectured to arise as an effective structure of the algebraic and dynamical relations between observables that are imposed by the background statistics of the system. Namely, we suggest that quantum reference states on an extended observable algebra, the free algebra generated by the observables, may give rise to effective spacetime structures. Accordingly, perturbations of the reference state lead to perturbations of the induced effective spacetime geometry. We initiate the study of these perturbations, and their relation to gravitational phenomena

The Principle of Locality. Effectiveness, fate and challenges

The Special Theory of Relativity and Quantum Mechanics merge in the key principle of Quantum Field Theory, the Principle of Locality. We review some examples of its ``unreasonable effectiveness'' (which shows up best in the formulation of Quantum Field Theory in terms of operator algebras of local observables) in digging out the roots of Global Gauge Invariance in the structure of the local observable quantities alone, at least for purely massive theories; but to deal with the Principle of Local Gauge Invariance is still a problem in this frame. This problem emerges also if one attempts to figure out the fate of the Principle of Locality in theories describing the gravitational forces between elementary particles as well. Spacetime should then acquire a quantum structure at the Planck scale, and the Principle of Locality is lost. It is a crucial open problem to unravel a replacement in such theories which is equally mathematically sharp and reduces to the Principle of Locality at larger scales. Besides exploring its fate, many challenges for the Principle of Locality remain; among them, the analysis of Superselection Structure and Statistics also in presence of massless particles, and to give a precise mathematical formulation to the Measurement Process in local and relativistic terms; for which we outline a qualitative scenario which avoids the EPR Paradox.Comment: 36 pages. Survey partially based on a talk delivered at the Meeting "Algebraic Quantum Field Theory: 50 years", Goettingen, July 29-31, 2009, in honor of Detlev Buchholz. Submitted to Journal of Mathematical Physic