IIIM 2018 – Incontro Italiano Insiemi e Modelli 2018
Data e luogo: 21-23 febbraio 2018 - Torino
Organizzatori:
Alessandro Andretta, Gianluca Basso, Riccardo Camerlo, Luca Motto Ros, Matteo Viale, Domenico Zambella
Luogo:
Dipartimento di Matematica
"Giuseppe Peano",
Palazzo Campana,
via Carlo Alberto 10, Torino.
Informazioni utili (in Inglese)
Programma
Giorno 1 - mercoledì 21 febbraio
- 09:00 - Registration
- 09:30 - 10:00 - Antongiulio Fornasiero
Espansioni di modelli di Presburger
Abstract.
Sia B una struttura intermedia tra un modello dell'aritmetica A con solo l'addizione ed uno con anche l'ordine. Se B è stabile, allora è uguale ad A.
Nascondi. - 10:05 - 10:55 - Lorna Gregory
Interpretation functors, representation type and decidability
Abstract.
In this talk I will give an overview of recent developments in the model theory of representations of finite-dimensional algebras.
These results will connect representation type with decidability of theories of modules and interpretation functors, an additive version of the model theoretic notion of interpretation.
The representation type of a finite-dimensional k-algebra is an algebraic measure of how hard it is to classify its finite-dimensional indecomposable modules. Roughly, a finite-dimensional k-algebra is of wild representation type if classifying its finite-dimensional indecomposable modules is as hard as classifying those of the polynomial ring in two non-commuting variables. On the other hand, a finite-dimensional algebra is tame if for every dimension d, all but finitely many of the finite-dimensional indecomposable modules of dimension d are in finitely many 1-parameter families. According to Drozd, when k is algebraically closed, a finite-dimensional k-algebra is either tame or wild.
Nascondi. - 10:55 – 11:20 - Coffee Break
- 11:20 - 11:50 - Silvia Barbina
Model theory of Steiner triple systems
Abstract.
Finite Steiner triple systems (STSs) are well known combinatorial objects for which the literature is extensive. An STS is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. The existence of the Fraïssé limit M of all finite STSs is known, but no description of its interesting theory was available so far. In joint work with Enrique Casanovas we describe the theory of M, which has quantifier elimination, TP2 and NSOP1.
Nascondi. - 11:55 - 12.25 - Sonia L'Innocente
Abelian regularization of rings and modules
Abstract.
This report aims at describing how Olivier's construction for the commutative regularization of a commutative ring can be generalized
to obtain the abelian regularization of an associative ring R. The Ziegler Spectrum of this abelian regularization and its relation with the constructible Cohn spectrum of R, (i.e., the Cohn spectrum equipped with the patch topology) are investigated. It is also shown how Prest's notion of the sheaf of definable scalars on the constructible Cohn spectrum generalizes Wiegand's description of the commutative regularization.
Nascondi. - 12:30 - 13.00 - Lorenzo Luperi Baglini
Nonstandard characterizations of partition regularity
Abstract.
In recent years, a nonstandard technique based on the characterization of monads of ultrafilters has been used to study several properties in Ramsey theory related with Diophantine equations. We will show how this technique can be adapted to study several other problems in combinatorics, including the characterization of arbitrary (finite) tensor products of ultrafilters. Iterated hyperextensions are one of the key ingredients of this study.
Nascondi. - 13:00 – 15:00 - Lunch
- 15:00 - 15:50 - Vassilis Gregoriades
Basis Results and the Spector-Gandy Theorem
Abstract.
A point $\beta \in \omega^\omega$ is a basis for a class $F$ of subsets of $\omega^\omega$ if every non-empty member of $F$ contains a point that is ``reasonably definable" from $\beta$. For example there is a specific $\beta \in \omega^\omega$ such that every recursive tree on the naturals with non-empty body contains a member, which is recursive in $\beta$.
It is natural to ask if there is a ``best basis" for a given class $F$ as above. In this talk we will give some examples of classes with best basis that are related to recursive metric spaces. The main tool for the proofs is the Spector-Gandy Theorem.
Nascondi. - 15:55 - 16:20 - Filippo Calderoni
The bi-embeddability relation between torsion-free abelian groups
Abstract.
Working in the framework of generalized descriptive set theory, we prove that the bi-embeddability relation on torsion-free abelian groups of cardinality $\kappa$, for $\kappa = \kappa^{<\kappa}$, is a complete analytic equivalence relation. This extends to uncountable structures a result obtained by the speaker and S. Thomas in the classical setting.
Nascondi. - 16:20 – 16:55 - Coffee Break
- 16:55 - 17:25 - Silvia Steila
When the reals form a proper class
Abstract.
In this talk we focus on some properties of real numbers over a conservative second order version of Kripke Platek Set Theory. In particular we focus on fixed-points statements and largeness properties of the class of real numbers in this setting. (on-going work with Gerhard Jäger)
Nascondi. - 17:30 - 18:00 - Vittorio Bard
Martin's conjecture and Borel monoid acts
Abstract.
Martin's conjecture is a long-lasting conjecture in recursion theory (regarding the endomorphisms of Turing equivalence). It was recently revived by the work of Simon Thomas, who showed that its assumption leads to tremendous consequences on the theory of countable Borel equivalence relations. A special case of Martin's conjecture was solved in the '80s by Slaman and Steel. In our talk, we introduce "monoid acts" (which are the analog of group actions, but with a monoid acting instead), as well as homomorphisms and reductions between Borel acts. Then, we show that Slaman and Steel's result amounts to Martin's conjecture for endomorphisms of the most natural Borel act generating Turing equivalence. Following Thomas' work, this result gives us important information on the structure of Borel acts of countable monoids, up to Borel reducibility.
Nascondi.
Giorno 2 - giovedì 22 febbraio
- 09:30 - 10:00 - Gianluca Paolini
Automorphisms Groups and Reconstruction Problems
Abstract.
We give an overview of recent results on the theory of automorphism groups of countable structures, centred around the problem of reconstruction of model-theoretic properties of a structure from the properties of its automorphism group. On the positive side, we prove a general sufficient condition for strong small index property (SSIP) and a new reconstruction up to bi-definability result for structures with SSIP, in analogy with Rubin's well-known result for structures admitting a $\forall\exists$-interpretation. On the negative side, we show that no topological or algebraic property of the group of automorphisms of a countable structure can detect any form of stability.
Nascondi. - 10:05 - 10:55 - David Aspero
Iterated forcing with side conditions and adding few new reals
Abstract.
There is a useful method for constructing models of set theory with the continuum arbitrarily large, involving iterated forcing with symmetric systems of models as side conditions. It turns out that a natural variation of this method can be used to produce forcing constructions giving rise to models of the Continuum Hypothesis. Unlike the case with the classical approaches to preserving CH, this new method produces forcing notions which actually add new reals, although only a small number of them. I will introduce this method, focusing on one specific application. Most of what I will present is joint work with M.A. Mota.
Nascondi. - 10:55 – 11:20 - Coffee Break
- 11:20 - 11:50 - Giorgio Laguzzi
Egalitarian and Paretian principles vs irregular sets
Abstract.
In the last years, some authors have investigated a certain connection between set theory and theoretical economics. In particular, Zame and Lawers have focused on Paretian pre-orderings on utility streams with infinite horizon, and found (in 2007 and 2009, respectively) interesting connections with irregular sets, such as non-Lebesgue measurable and non-Ramsey sets. In this talk we prove some further results about this kind of interplays, in particular involving the Baire property and other types of social welfare relations satisfying egalitarian principles, such as Hammond's equity.
Nascondi. - 11:55 - 12:25 - Giorgio Audrito
Normal systems of filters
Abstract.
- 12:30 - 13:00 - Philipp Moritz Lücke
Simply definable failures of weak compactness
Abstract.
A result of Hung and Negrepontis shows that an uncountable regular cardinal $\kappa$ is weakly compact if and only if the space ${}^\kappa\kappa$, consisting of all functions from $\kappa$ to $\kappa$ equipped with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $\kappa$, is not homeomorphic to the subspace ${}^\kappa 2$ of ${}^\kappa\kappa$ consisting of all binary functions. Motivated by recent work of Andretta and Motto Ros, we consider the question whether homeomorphisms of ${}^\kappa\kappa$ and ${}^\kappa 2$ witnessing failures of weak compactness can be simply definable. Our results show that both large cardinals axioms and forcing axioms imply that homomorphisms witnessing that $\omega_1$ is not weakly compact are not simply definable. In contrast, we show that the analogous statements for $\omega_2$ can consistently fail. Finally, we also consider the above questions for successors of singular cardinals and inaccessible cardinals that are not weakly compact.
Nascondi. - 13:00 – 15:00 - Lunch
- 14:25 – 14:55 - Discussion session: G. Basso e F. Damiani
- 15:00 - 15:50 - Mauro Di Nasso
Nonstandard natural numbers in combinatorics
and Ramsey Theory
Abstract.
In the last years, the methods of nonstandard analysis
have been applied to combinatorial number theory producing
several new results. In the first part of my talk I will
give a brief introduction to the use of nonstandard integers
in the area of combinatorics where one studies structural
properties of sets of integers that only depend on their
positive asymptotic density. I will then show how nonstandard
integers can play the role of ultrafilters and be used in Ramsey theory.
Finally, I will present some new results obtained by applying
the presented nonstandard methods.
Nascondi. - 15:55 - 16:25 - Carlo Toffalori
The torsion free part of the Ziegler spectrum of orders over Dedekind domains
Abstract.
We introduce and illustrate recent results obtained with Lorna Gregory and Sonia L'Innocente on the model theory of modules over orders over Dedekind domains. They enlarge a past analysis over group rings. Here orders are meant as certain widely considered finitely generated ring extensions of Dedekind domains.
Nascondi. - 16:25 – 16:55 - Coffee Break
- 16:55 - 17:25 - Francesco Parente
On regular ultrafilters, Boolean ultrapowers, and Keisler's order
Abstract.
In this talk, we shall present some applications of the Boolean ultrapower construction to the study of Keisler's order.
Over the last decade, Malliaris and Shelah proved a striking sequence of results in the intersection between model theory and set theory, settled affirmatively the question of whether $\mathfrak{p}=\mathfrak{t}$, and developed surprising connections between classification theory and cardinal characteristics of the continuum. The main motivation of their work is the study of Keisler's order, introduced originally in 1967 as a device to compare the complexity of complete theories by looking at regular ultrapowers of their models.
In this context, there has been a recent shift towards building ultrafilters on complete Boolean algebras. In particular, moral ultrafilters have emerged as the main tool to find dividing lines among unstable theories.
Motivated by this new Boolean-algebraic framework, in the first part of the talk we shall analyse and compare two different notions of regularity for ultrafilters on complete Boolean algebras. This analysis will show that most model-theoretic properties of $\kappa$-regular ultrafilters can be generalized smoothly to the context of $(\kappa,2)$-distributive Boolean algebras. On the other hand, we shall prove the existence of regular ultrafilters on the Cohen algebra $\mathbb{C}_\kappa$ with unexpected model-theoretic features.
In the second part of the talk, the following question will be addressed: what kind of classification can arise when we compare theories according to the saturation of Boolean ultrapowers of their models? In order to provide an answer to this question, we shall introduce a new Boolean-algebraic analogue of Keisler's order and compare it with the usual one.
Nascondi.
- Cena Sociale al Ristorante Quadre (map)
Menù.
- Antipasto: Flan di carote con le sue chips e fonduta di caprino
- Piatto principale:
- – Gulash alla paprika dolce con broccoli profumati al limone e patate al forno
- – Cipolla ripiena di crema di ceci e pomodori secchi, su crema di patate e porri (vegetariani)
- Dolce: Tortino di nocciole con crema di pere
Nascondi menù.
Giorno 3 - venerdì 23 febbraio
- 09:30 - 10:00 - Alessandro Vignati
Triviality and nontriviality of homeomorphisms of Cech-Stone remainders
Abstract.
Given a locally compact space $X$, we define an homeomorphism of its Cech-Stone remainder $\beta X \setminus X$ to be trivial if it is induced by an homeomorphism between cocompact subsets of $X$. For example, if $X=\omega$ trivial homeomorphisms correspond to almost permutations on omega. Are all homeomorphisms of $\beta X \setminus X$ trivial? Rudin, Shelah, and Velickovic showed that in case $X=\omega$ the answer to this question depends on set theory. We report on recent successful attempts to extend this intuition to more general locally compact spaces.
Nascondi. - 10:05 - 10:55 - Alessandro Berarducci
Homotopy, dimension and hyperdefinability
Abstract.
Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ with strong connectivity conditions on the fibers is given. We apply similar techniques in o-minimal expansions of fields to compare the o-minimal homotopy of a definable set $X$ with the classical homotopy of some of its bounded hyperdefinable quotients $X/E$. Although the homotopy type of a space does not determine its dimension, under suitable hypothesis we obtain similar transfer results for the dimension. As a special case, given a definably compact group, we obtain a new proof of Pillay's conjecture "$\dim(G)=\dim_{\mathbb R}(G/G_{00})$" largely independent of the group structure of $G$. We also obtain as special cases various transfer results by Delfs-Knebush and Baro-Otero relating the o-minimal and the classical homotopy of an o-minimal space.
Nascondi. - 10:55 – 11:20 - Coffee Break
- 11:20 - 11:50 - Philipp Schlicht
Class forcing and reverse mathematics of second-order set theory
Abstract.
The forcing theorem, the most fundamental result about set forcing, states that every set forcing admits forcing relations for all formulas and the truth lemma holds: statements true in the corresponding forcing extensions are forced and forced statements are true. Unlike for set forcing, the forcing theorem for class forcing can fail in models of Gödel-Bernays class theory GBC with global choice. We show that the class forcing theorem is equivalent over the base theory GBC to the principle ETR_Ord of class recursion of length Ord; the existence of Ord-length iterated truth predicates relative to any class; the statement that every separative class partial order has a set-complete class Boolean completion; and the principle of determinacy for clopen class games of rank at most Ord+1. This situates the class forcing theorem in an emerging hierarchy of reverse mathematics of second-order set theory. This is joint work with Victoria Gitman, Joel Hamkins, Peter Holy and Kameryn Williams.
Nascondi. - 11:55 - 11:25 - Marco Forti
Euclidean transfinite sums of integers and Euclidean numerosity of sets
Abstract.
- 12:30 - 13:00 - Alberto Marcone
When embeddability and epimorphism agree: strongly surjective linear orders
Abstract.
A linear order L is strongly surjective if there exists an order
preserving surjection from L onto each of its suborders.
Our main result is that the set StS of countable strongly surjective
linear orders is the union of an analytic and a coanalytic set, and is
complete for the class of sets that can be written in this way. More in
detail, we show that the countable strongly surjective linear orders
which are scattered form a coanalytic-complete set, while the
countable strongly surjective linear orders which are not scattered form
an analytic-complete set.
Even if the study of the first two levels of the projective hierarchy is
a long-standing topic, examples of sets that are true Delta^1_2 are very
rare. In fact, as far as we know, StS is the first example of a
"natural" set which is complete for the class of unions of an analytic
and a coanalytic set.
This is joint work with Riccardo Camerlo and Raphael Carroy
Nascondi.
Partecipanti registrati
Alessando Andretta, Università di Torino.
David Aspero, University of East Anglia.
Giorgio Audrito, Università di Torino.
Silvia Barbina, Open University.
Vittorio Bard, Università di Torino.
Gianluca Basso, Università di Torino, Université de Lausanne.
Alessandro Berarducci, Università di Pisa.
Filippo Calderoni, Università di Torino.
Riccardo Camerlo, Politecnico di Torino.
Filippo Cavallari, Università di Torino - Université de Lausanne.
Fernando Damiani, Università degli Studi di Roma “La Sapienza”.
Mauro Di Nasso, Università di Pisa.
Vincenzo Dimonte, Università degli Studi di Udine.
Antongiulio Fornasiero, Università di Firenze.
Marco Forti, Università di Pisa.
Vassilis Gregoriades, Università di Torino.
Lorna Gregory, Università di Camerino.
Sonia L'Innocente, Università di Camerino.
Giorgio Laguzzi, University of Freiburg.
Philipp Moritz Lücke, Mathematisches Institut der Universität Bonn.
Lorenzo Luperi Baglini, University of Vienna.
Marcello Mamino, TU Dresden.
Francesco Mangraviti, University Claude Bernard Lyon 1.
Alberto Marcone, Università di Udine.
Luca Motto Ros, Università di Torino.
Gianluca Paolini, Hebrew University of Jerusalem.
Francesco Parente, University of East Anglia.
Philipp Schlicht, Universität Bonn.
Silvia Steila, Universität Bern.
Giuseppina Terzo, Università degli Studi della Campania "L. Vanvitelli".
Carlo Toffalori, Università di Camerino.
Andrea Vaccaro, Università di Pisa & York University.
Manlio Valenti, Università degli studi di Udine.
Matteo Viale, Università di Torino.
Alessandro Vignati, IMJ-PRG - Universite Paris 7.
Domenico Zambella, Università di Torino.
Sponsors
L'incontro è generosamente sostenuto da:
- Progetto PRIN 2012 "Modelli e insiemi" (PI: Carlo Toffalori)
- Dipartimento di Matematica G. Peano, Torino