Finding a field in a Zariski-like structure

The starting point for this dissertation is whether the concept of Zariski geometry, introduced by Hrushovski and Zilber, could be generalized to the context of non-elementary classes. This leads to the axiomatization of Zariski-like structures. As our main result, we prove that if the canonical pregeometry of a Zariski-like structure is non locally modular, then the structure interprets either an algebraically closed field or a non-classical group. This is a counterpart to the result by Hrushovski and Zilber which states that an algebraically closed field can be found in a non locally modular Zariski geometry. It demonstrates that the concept of a Zariski-like structure captures one of the most essential features of a Zariski geometry. Finally, we give a non-trivial example by showing that the cover of the multiplicative group of an algebraically closed field of characteristic zero is Zariski-like. We define a Zariski-like structure as a quasiminimal pregeometry structure that has certain properties. Instead of assuming underlying topologies, we formulate the axioms for a countable collection C of Galois definable sets that have some of the properties of irreducible closed sets from the Zariski geometry context. Quasiminimal classes are abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. They are uncountably categorical, and have both the amalgamation property (AP) and the joint embedding property (JEP), and thus also a model homogeneous universal monster model, which we denote by M. To adapt Hrushovski's and Zilber's proof to our setting, we first generalize Hrushovski's Group Configuration Theorem to the context of quasiminimal classes. For this, we develop an independence calculus that has all the usual properties of non-forking and works in our context. We then prove the group configuration theorem and apply it to find a 1-dimensional group, assuming that the canonical pregeometry obtained from the bounded closure operator is non-trivial. A field can be found under the further assumptions that M does not interpret a non-classical group and the canonical pregeometry is non locally modular. Finally, we show that the cover of the multiplicative group of an algebraically closed field, studied by e.g. Boris Zilber and Lucy Burton, provides a non-trivial example of a Zariski-like structure. Burton obtained a topology on the cover by taking sets definable by positive, quantifier-free first order formulae as the basic closed sets. This is called the PQF-topology, and the sets that are closed with respect to it are called the PQF-closed sets. We show that the cover becomes Zariski-like after adding names for a countable number of elements to the language. The axioms for a Zariski-like structure are then satisfied if the collection C is taken to consist of the PQF-closed sets that are definable over the empty set.Väitöskirjani sijoittuu kahden matematiikan osa-alueen, algebrallisen geometrian ja matemaattisen logiikan piiriin kuuluvan malliteorian risteyskohtaan. Klassista geometriaa voi hahmottaa kuvien kautta, mutta sitä voi käsitellä myös algebrallisesti, esimerkiksi yhtälöiden avulla. Algebrallisessa geometriassa tällainen lähestymistapa on ensisijainen ja geometriset kappaleet määritellään algebrallisesti. Tämä tuottaa myös sellaisia geometrisia olioita, joista ei voi piirtää kuvaa. Niitä voidaan kuitenkin käsitellä algebran keinoin samalla tavalla kuin klassisempia geometrisia olioita. Yleensä algebrallisen geometrian olioilla on kiinteä yhteys kunnaksi nimitettyyn algebralliseen rakenteeseen. Malliteoria puolestaan tutkii ja luokittelee matemaattisia struktuureja, eli malleja. Malli on joukko, johon on määritelty rakenne jonkin formaalin kielen avulla. Yksi algebrallisen geometrian keskeisiä käsitteitä on Zariski-topologia, kokoelma joukkoja, joka tarjoaa välineistön geometristen olioiden käsittelyyn. Nämä joukot määritellään algebrallisesti. 1990-luvun alussa Ehud Hrushovski ja Boris Zilber lähestyivät algebrallista geometriaa malliteorian kautta ja kehittivät Zariski-geometrian käsitteen. Se on Zariski-topologiaa yleistävä malliteoreettinen struktuuri. Myös Zariski-geometrioissa tietty kokoelma joukkoja on olennaisessa roolissa, mutta sillä ei välttämättä ole suoraa yhteyttä algebraan. Hrushovski ja Zilber kuitenkin osoittivat, että tietyt geometriset ehdot täyttävästä Zariski-geometriasta voidaan aina löytää kunta. Algebrallisesta rakenteesta ei siis päästä eroon. Hrushovski hyödynsi Zariski-geometrioita todistaessaan algebrallista geometriaa ja lukuteoriaa koskevan geometrisen Mordell-Lang -konjektuurin (1996). Lisäksi Zilber on esittänyt, että Zariski-geometrioista (tai niiden tapaisista objekteista) voisi olla hyötyä rakennettaessa matemaattisesti vedenpitävää pohjaa kvanttikenttäteorialle. Zariski-geometriat on määritelty klassisessa malliteoriassa käytetyn formaalin kielen puitteissa. On kuitenkin olemassa monia mielenkiintoisia struktuureja, joiden kuvaamiseen tämän kielen ilmaisuvoima ei riitä. Tästä syystä onkin kehitetty abstraktimpi lähestymistapa, jossa malleja tarkastellaan pikemminkin niiden välisten suhteiden kuin niitä kuvaavan kielen kautta. Väitöskirjani lähtökohtana oli kysymys siitä, voidaanko Zariski-geometriat yleistää tähän abstraktimpaan kontekstiin. Työssä esitellään tällainen yleistys ja osoitetaan että algebra on siinäkin läsnä: kun tietyt geometriset ehdot täyttyvät, löytyy joko kunta tai ns. epäklassinen ryhmä. Toistaiseksi ei tiedetä, onko epäklassisia ryhmiä olemassa, mutta Hrushovskin ja Zilberin Zariski-geometrioissa niitä ei voi esiintyä. Väitöskirjassa annetaan myös epätriviaali esimerkki tällaisesta yleisemmästä struktuurista

Finding a field in a Zariski-like structure

We show that if M is a Zariski-like structure (see [7]) and the canonical pregeometry obtained from the bounded closure operator (bcl) is non-locally modular, then M interprets either an algebraically closed field or a non-classical group. (C) 2017 Elsevier B.V. All rights reserved.Peer reviewe

On model theory of covers of algebraically closed fields

We study covers of the multiplicative group of an algebraically closed field as quasiminimal pregeometry structures and prove that they satisfy the axioms for Zariski-like structures presented in \cite{lisuriart}, section 4. These axioms are intended to generalize the concept of a Zariski geometry into a non-elementary context. In the axiomatization, it is required that for a structure \M, there is, for each $n$, a collection of subsets of \M^n, that we call the \emph{irreducible sets}, satisfying certain properties. These conditions are generalizations of some qualities of irreducible closed sets in the Zariski geometry context. They state that some basic properties of closed sets (in the Zariski geometry context) are satisfied and that specializations behave nicely enough. They also ensure that there are some traces of Compactness even though we are working in a non-elementary context

Families of abelian varieties with many isogenous fibres

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Gauge theory in dimension 77

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Relative Property (T) and Linear Groups

Relative property (T) has recently been used to construct a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group Gamma admits a real-special linear representation with non-amenable Zariski closure if and only if it acts on an Abelian group A (of finite nonzero Q-rank) so that the corresponding group pair (Gamma \ltimes A, A) has relative property (T). The proof is constructive. The main ingredients are Furstenberg's celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.Comment: 34 pages, last section is ne