Advanced seminar
Upcoming:
Past:

January 18th, 2018, 14.0015.30 (Palazzo Campana, Aula 3)
V. Bard (Torino) "Martin's conjecture and Borel acts", part 1.
Martin's conjecture is probably one of the most famous open problems in Turing degree theory, as well as the only Victoria Delfino problem (see Cabal Seminar 7677) which remains open to this day. The idea behind its formulation is that, despite the general complexity of the structure of Turing degrees, if one limits himself with those Turing degrees which are commonly found "in nature", then one sees a very simple structure. The mathematical precise notion that the statement of Martin's conjecture uses to capture the notion of "natural Turing degree" is that of "endomorphism of Turing equivalence in a model of AD". In the 80's, Slaman and Steel proved Martin's conjecture for a particular class of endomorphisms of Turing equivalence; we will show that this class coincides with the class of endomorphisms of the natural (partial) act generating Turing equivalence. An "act" is the analog of a group action, but with a monoid acting instead; a particual kind of acts, the countable Borel ones, are crucial in the theory of countable Borel quasiorders (such as Turing reducibility) because of a FeldmanMoorelike theorem: countable Borel quasiorders are exactly those quasiorders induced by countable Borel acts. So, we will discuss how much the proof of Slaman and Steel can be extended to endomorphisms of different Borel acts generating Turing equivalence, and thus how close we can get to a proof of Martin's conjecture via this approach. Moreover, Martin's conjecture has turned out to be related with the Borel cardinality of Turing equivalence, and with the question: "How many weakly universal countable Borel equivalence relations are there (up to Borel bireducibility)?". So, we will consider the notion of Borel reducibility between Borel acts, as well as a number of strengthenings of that notion (one of which is particularly interesting because of its ubiquity in universality proofs of countable Borel equivalence relations), and we will show that Slaman's and Steel's result that Martin's conjecture holds for the endomorphisms of the Turing act allows to prove some interesting facts about universal countable Borel acts.
Hide abstract. 
February 1st, 2018, 14.0015.30 (Palazzo Campana, Aula 1)
V. Bard (Torino) "Martin's conjecture and Borel acts", part 2.
Martin's conjecture is probably one of the most famous open problems in Turing degree theory, as well as the only Victoria Delfino problem (see Cabal Seminar 7677) which remains open to this day. The idea behind its formulation is that, despite the general complexity of the structure of Turing degrees, if one limits himself with those Turing degrees which are commonly found "in nature", then one sees a very simple structure. The mathematical precise notion that the statement of Martin's conjecture uses to capture the notion of "natural Turing degree" is that of "endomorphism of Turing equivalence in a model of AD". In the 80's, Slaman and Steel proved Martin's conjecture for a particular class of endomorphisms of Turing equivalence; we will show that this class coincides with the class of endomorphisms of the natural (partial) act generating Turing equivalence. An "act" is the analog of a group action, but with a monoid acting instead; a particual kind of acts, the countable Borel ones, are crucial in the theory of countable Borel quasiorders (such as Turing reducibility) because of a FeldmanMoorelike theorem: countable Borel quasiorders are exactly those quasiorders induced by countable Borel acts. So, we will discuss how much the proof of Slaman and Steel can be extended to endomorphisms of different Borel acts generating Turing equivalence, and thus how close we can get to a proof of Martin's conjecture via this approach. Moreover, Martin's conjecture has turned out to be related with the Borel cardinality of Turing equivalence, and with the question: "How many weakly universal countable Borel equivalence relations are there (up to Borel bireducibility)?". So, we will consider the notion of Borel reducibility between Borel acts, as well as a number of strengthenings of that notion (one of which is particularly interesting because of its ubiquity in universality proofs of countable Borel equivalence relations), and we will show that Slaman's and Steel's result that Martin's conjecture holds for the endomorphisms of the Turing act allows to prove some interesting facts about universal countable Borel acts.
Hide abstract. 
February 16th, 2018, 10.3012.00 (Palazzo Campana, Aula 3)
V. Bard (Torino) "Martin's conjecture and Borel acts", part 3.
Martin's conjecture is probably one of the most famous open problems in Turing degree theory, as well as the only Victoria Delfino problem (see Cabal Seminar 7677) which remains open to this day. The idea behind its formulation is that, despite the general complexity of the structure of Turing degrees, if one limits himself with those Turing degrees which are commonly found "in nature", then one sees a very simple structure. The mathematical precise notion that the statement of Martin's conjecture uses to capture the notion of "natural Turing degree" is that of "endomorphism of Turing equivalence in a model of AD". In the 80's, Slaman and Steel proved Martin's conjecture for a particular class of endomorphisms of Turing equivalence; we will show that this class coincides with the class of endomorphisms of the natural (partial) act generating Turing equivalence. An "act" is the analog of a group action, but with a monoid acting instead; a particual kind of acts, the countable Borel ones, are crucial in the theory of countable Borel quasiorders (such as Turing reducibility) because of a FeldmanMoorelike theorem: countable Borel quasiorders are exactly those quasiorders induced by countable Borel acts. So, we will discuss how much the proof of Slaman and Steel can be extended to endomorphisms of different Borel acts generating Turing equivalence, and thus how close we can get to a proof of Martin's conjecture via this approach. Moreover, Martin's conjecture has turned out to be related with the Borel cardinality of Turing equivalence, and with the question: "How many weakly universal countable Borel equivalence relations are there (up to Borel bireducibility)?". So, we will consider the notion of Borel reducibility between Borel acts, as well as a number of strengthenings of that notion (one of which is particularly interesting because of its ubiquity in universality proofs of countable Borel equivalence relations), and we will show that Slaman's and Steel's result that Martin's conjecture holds for the endomorphisms of the Turing act allows to prove some interesting facts about universal countable Borel acts.
Hide abstract. 
March 23rd, 2018, 10.3012.30 (Palazzo Campana, Aula Lagrange)
R. Treglia (Torino) "Introduction to $\lambda$calculus as a term rewriting system", part 1.
The $\lambda$calculus is a collection of formal theories whose definitions are given by three constructors and a single computational rule, namely the $\beta$reduction. $\lambda$calculus originated from certain systems of combinatory logic that were originally proposed as a foundation of mathematics around 1930 by Church and Curry. Those systems were subsequently shown to be inconsistent by Church's students Kleene and Rosser in 1935, leading to a negative answer to Hilbert's Entscheidungsproblem, but a certain subsystem consisting of the $\lambda$terms equipped with socalled $\beta$reduction turned out to be useful in formalizing the intuitive notion of effective computability and led to Church's thesis. Indeed, $\lambda$calculus is an appropriate formalization of the intuitive notion of effective computability. During the first seminar, we will focus on drawing the history of the concepts that led to the first formulation by Alonzo Church in 1928. Then, Kleene and Rosser paradox will be dealt as an semantic paradox that afflicts the first formalization. After the first calculus inconsistency was proved, Church published a revised calculus, less powerful but correct. Before introducing the revised $\lambda$calculus with the maximum power of its $\beta$reduction, a simpler and deterministic dialect is shown. All definitions will be formulated in the rewriting theory. At the end of the seminar a fundamental theorem in rewriting theory, the ChurchRosser theorem, will be enunciated.
Hide abstract. 
April 5th, 2018, 14.3016.30 (Palazzo Campana, Aula 3)
R. Carroy (Vienna) "Wadge theory and an application to homogeneous spaces".
I will start by giving an overview of a possible analysis of the Wadge theory. Fons van Engelen famously used the description of Wadge degrees of Borel sets to analyze Borel homogeneous spaces. I will explain the first steps we have made with Andrea Medini and Sandra Müller towards the generalization of van Engelen's results in the projective hierarchy.
Hide abstract. 
April 6th, 2018, 11.3013.30 (Palazzo Campana, Aula Lagrange)
R. Treglia *CANCELLED* (Torino) "
Introduction to $\lambda$calculus as a term rewriting system", part 2.The $\lambda$calculus is a collection of formal theories whose definitions are given by three constructors and a single computational rule, namely the $\beta$reduction. $\lambda$calculus originated from certain systems of combinatory logic that were originally proposed as a foundation of mathematics around 1930 by Church and Curry. Those systems were subsequently shown to be inconsistent by Church's students Kleene and Rosser in 1935, leading to a negative answer to Hilbert's Entscheidungsproblem, but a certain subsystem consisting of the $\lambda$terms equipped with socalled $\beta$reduction turned out to be useful in formalizing the intuitive notion of effective computability and led to Church's thesis. Indeed, $\lambda$calculus is an appropriate formalization of the intuitive notion of effective computability. During the first seminar, we will focus on drawing the history of the concepts that led to the first formulation by Alonzo Church in 1928. Then, Kleene and Rosser paradox will be dealt as an semantic paradox that afflicts the first formalization. After the first calculus inconsistency was proved, Church published a revised calculus, less powerful but correct. Before introducing the revised $\lambda$calculus with the maximum power of its $\beta$reduction, a simpler and deterministic dialect is shown. All definitions will be formulated in the rewriting theory. At the end of the seminar a fundamental theorem in rewriting theory, the ChurchRosser theorem, will be enunciated.
Hide abstract. 
April 6th, 2018, 14.3016.30 (Palazzo Campana, Aula 5)
S. Müller (Vienna) "Combinatorial Variants of Lebesgue's Density Theorem".
Lebesgue introduced a notion of density point of a set of reals and proved that any Borel set of reals has the density property, i.e. it is equal to the set of its density points up to a null set. We introduce alternative definitions of density points in Cantor space (or Baire space) which coincide with the usual definition of density points for the uniform measure on ${}^{\omega}2$ up to a set of measure 0, and which depend only on the ideal of measure 0 sets but not on the measure itself. This allows us to define the density property for the ideals associated to tree forcings analogous to the Lebesgue density theorem for the uniform measure on ${}^{\omega}2$. The main results show that among the ideals associated to wellknown tree forcings, the density property holds for all such ccc forcings and fails for the remaining forcings. In fact we introduce the notion of being stemlinked and show that every stemlinked tree forcing has the density property. This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.
Hide abstract. 
April 20th, 2018, 10.3012.30 (Palazzo Campana, Aula Lagrange)
R. Treglia (Torino) "$\lambda$calculus and abstract machines ", part 2.
This seminar deals with the study of time cost models for $\lambda$calculus. Even the simplest dialects of the $\lambda$calculus are affected by a phenomenon called size explosion where the $\beta$reduction steps do not seem to be a metric for the complexity of a given calculus. Despite the above mentioned difficulty, it turns out that one can actually use $\beta$steps to analyse the complexity of a specific evaluation strategy. In the first part of the seminar the focal point is the connection between $\lambda$calculus and Turing Machines. From a historical point of view this connection leads to the ChurchTuring thesis, but from our point of view it gives the possibility to shift the attention from the formal language to the machines that mimic it. The main topics of the second part are abstract machine. By means of them, that are implementation schemas for fixed calculi that are a compromise between theory and practice: they are concrete enough to provide a notion of machine, and abstract enough to avoid the many intricacies of actual implementations. Abstract machines can be used to prove that the number of $\beta$steps is a reasonable time cost model, i.e. a metric for time complexity. The correspondence between an abstract machine and its associate calculus is usually proved via suitable implementation theorems, which ensure that there is a perfect matching between a machine and the respective strategy. This will guarantee that in order to estimate the complexity of the strategy it will be enough to study the overall complexity of the corresponding machine.
Hide abstract. 
April 27th, 2018, 10.3012.30 (Palazzo Campana, Aula Lagrange)
V. Giambrone (Torino) "Boolean valued models for Set Theory and Grothendieck Topoi", part 1.
In this talk we will present some aspects of the connection between boolean valued models for Set Theory (a subject pertaining to logic and Set Theory) with sheaves and topoi (which are mostly studied by category theorists and algebraic geometers). Boolean valued models for Set Theory are a standard method to present Forcing. The forcing technique was introduced by Cohen in 1963 in order to prove the independence of the Continuum Hypotesis from the ZFC axioms for Set Theory. Since then it has been applied to prove the undecidability of many problems arising in various branches of mathematics, among others: group theory, topology, functional analysis. Category Theory arose from a 1945 article written by Mac Lane and Eilenberg on algebraic topology. Its high degree of abstraction allows to find applications of category theoretic ideas and methods almost everywhere in mathematics. Even if the idea of dealing with forcing from a categorical point of view has been well developed, the interpretation of boolean valued models for Set Theory as categories of sheaves on a boolean topological space has not been explored in full details yet, and we will make a first step towards this aim.
Hide abstract. 
May 4th, 2018, 10.3012.30 (Palazzo Campana, Aula 2)
R. Treglia *CANCELLED* (Torino) "
TBA", part 3. 
May 18th, 2018, 10.3012.30 (Palazzo Campana, Aula Lagrange)
V. Giambrone (Torino) "Boolean valued models for Set Theory and Grothendieck Topoi", part 2.
In this talk we will present some aspects of the connection between boolean valued models for Set Theory (a subject pertaining to logic and Set Theory) with sheaves and topoi (which are mostly studied by category theorists and algebraic geometers). Boolean valued models for Set Theory are a standard method to present Forcing. The forcing technique was introduced by Cohen in 1963 in order to prove the independence of the Continuum Hypotesis from the ZFC axioms for Set Theory. Since then it has been applied to prove the undecidability of many problems arising in various branches of mathematics, among others: group theory, topology, functional analysis. Category Theory arose from a 1945 article written by Mac Lane and Eilenberg on algebraic topology. Its high degree of abstraction allows to find applications of category theoretic ideas and methods almost everywhere in mathematics. Even if the idea of dealing with forcing from a categorical point of view has been well developed, the interpretation of boolean valued models for Set Theory as categories of sheaves on a boolean topological space has not been explored in full details yet, and we will make a first step towards this aim.
Hide abstract. 
May 30th, 2018, 09.3011.00 (Aula C)
L. Spada (Salerno) "Dualità categoriali".
Le dualità categoriali sono risultati matematici che stabiliscono profondi legami tra classi di oggetti in generale molto diverse. Ad esempio, la dualità di Stone, che stabilisce che le algebre di Boole sono dualmente equivalenti agli spazi compatti di Hausdorff zero dimensionali, è stato il primo risultato a mostrare la vicinanza tra due temi considerati distanti: algebra e topologia. Il seminario sarà un'occasione per inquadrare in un ambito matematico generale alcune dualità (Stone, Priestley, Gelfand, etc.), di importanza in logica e in matematica, e introdurre il linguaggio categoriale che ne permette una formulazione rigorosa.
Hide abstract. 
May 31st, 2018, 14.3015.30 (Politecnico, Aula 1D)
G. Basso (Lausanne and Torino) "Finding universal compact spaces, with some help from logic.".
We use Fraïssé limits, a well known construction from mathematical logic, to find compact metric spaces which have projective universality and homogeneity properties with respect to a given class of spaces and maps. To this end we encode the topological information of a compact space by a projective sequence of finite graphs and morphisms between them. When a projective sequence respects some combinatorial properties with respect to a given family of structures its limit enjoys universality and homogeneity properties which can be transferred to the compact space which is coded by the sequence, together with dynamical information on its group of homeomorphisms. This approach was developed by Irwin and Solecki in 2007 to investigate the pseudoarc, an interesting indecomposable continuum, and has since constituted a fertile field of research.
Hide abstract. 
June 1st, 2018, 10.3012.30 (Aula A)
L. Spada (Salerno) "Kakutani duality for groups".
Let I be any set, one can define a "denominator" function on [0,1]^I, by sending each point in [0,1]^I \ Q^I to 0 and otherwise to the least common multiple of the denominators of the coordinates, written in reduced form, (the lcm being 0, in case the set of denominators is unbounded). Suppose that the points of a compact Hausdorff space X are labeled with natural numbers by a function d: X>N . When does there exist an embedding of X into [0,1]^I, for some set I, that preserves d? By "preserving d" here we mean that points labeled by d with a natural number n go into points with denominator equal to n. A “reasonable” solution to the above problem gives a “reasonable” description of the category which is dual to normcomplete lattice ordered groups, thereby extending Kakutani duality for norm complete vector lattices.
Hide abstract.

January 11th, 2017, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino) "Classes of Polish spaces under effective Borel isomorphism", part 7.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract. 
January 18th, 2017, 11.3013.00 (Palazzo Campana, Aula 2)
V. Gregoriades (Torino) "Classes of Polish spaces under effective Borel isomorphism", part 8.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract. 
January 25th, 2017, 11.3013.00 (Palazzo Campana, Sala S)
V. Gregoriades (Torino) "Classes of Polish spaces under effective Borel isomorphism", part 9.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract. 
February 1st, 2017, 11.3013.00 (Palazzo Campana, Aula 2)
V. Gregoriades (Torino) "Classes of Polish spaces under effective Borel isomorphism", part 10.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract. 
February 8th, 2017, 11.3013.00 (Palazzo Campana, Sala S)
J. Somaglia (Milano e Praga) "Relations between coarse wedge topology on trees and retractional skeletons".
I will introduce the classes of Valdivia and noncommutative Valdivia compacta. After that I will recall the definition and some properties of the Coarse wedge topology on trees. Finally a characterization of noncommutative Valdivia trees will be presented.
Hide abstract. 
February 15th, 2017, 11.3013.00 (Palazzo Campana, Aula C)
R. Carroy (Vienna) "A dichotomy for topological embedding between functions", part 1.
A function f embeds in a function g when there are two topological embeddings a and b such that af=gb. I will prove that given any two Polish 0dimensional spaces X and Y this quasiorder is either analytic complete or a better quasiorder. This is a joint work with Yann Pequignot and Zoltan Vidnyanszky.
Hide abstract. 
February 22nd, 2017, 11.3013.00 (Palazzo Campana, Aula C)
R. Carroy (Vienna) "A dichotomy for topological embedding between functions", part 2.
A function f embeds in a function g when there are two topological embeddings a and b such that af=gb. I will prove that given any two Polish 0dimensional spaces X and Y this quasiorder is either analytic complete or a better quasiorder. This is a joint work with Yann Pequignot and Zoltan Vidnyanszky.
Hide abstract. 
March 7th, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
V. Dimonte (Udine) "Rankintorank axioms and forcing", part 1.
At the beginning of the development of rankintorank axioms, forcing did not have an important role, as such axioms are mostly left untouched by it, or completely destroyed. Recently a third way has appeared: an analysis of the structure of special sets under $I0$ lead to a sort of Generic Absoluteness Theorem, that implies many consistency result but that yields also concrete results, like the analogous of the Perfect Set Theorem.
Hide abstract. 
March 14th, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
V. Dimonte (Udine) "Rankintorank axioms and forcing", part 2.
At the beginning of the development of rankintorank axioms, forcing did not have an important role, as such axioms are mostly left untouched by it, or completely destroyed. Recently a third way has appeared: an analysis of the structure of special sets under $I0$ lead to a sort of Generic Absoluteness Theorem, that implies many consistency result but that yields also concrete results, like the analogous of the Perfect Set Theorem.
Hide abstract. 
March 17th, 2017, 10.3012.00 (Palazzo Campana, Aula 2)
F. Cavallari (Torino) "An Overview on Automata Theory ".
This is a basic seminar on Automata Theory. The goal of this seminar is to provide an overview of some basic notions like regular languages, parity automata, monadic second order logic, connections between Descriptive Set Theory and Automata, to prepare the audience for the seminar that Michał Skrzypczak (University of Warsaw) will give on the 21st of March.
Hide abstract. 
March 21st, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
M. Skrzypczak (Warsaw) "Connecting decidability and complexity for MSO logic".
During my presentation I will discuss connections between decidability and complexity. I will focus on Monadic SecondOrder (MSO) logic and its variants. On the ``decidability'' side, I will present standard and less standard results proving (un)decidability of this logic over some structures. On the ``complexity'' side, I will relate the decidability results to certain complexity measures.
Hide abstract.
The first of the complexity measures is the topological complexity of sets that can be defined in the given logic. In that case, it turns out that there are strong connections between high topological complexity of sets available in a given logic, and its undecidability. One of the milestone results in this context is the Shelah's proof of undecidability of MSO over reals.
The second complexity measure focuses on the mathematical strength needed to actually prove decidability of the given theory. The idea is to apply techniques of the reversed mathematics to the classical decidability results from automata theory. Recently, both crucial theorems of the area (the results of Buchi and Rabin) have been characterised in these terms. In both cases the proof gives strong relations between decidability of the MSO theory with other mathematical concepts: determinacy, Ramsey theorems, weak Konig's lemma, etc... 
March 28th, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
G. Basso (Lausanne and Torino) "Projective Fraïssé Limits of Partial Orders", part 1.
The KechrisPestovTodorčević correspondence links Ramsey Theory, Fraïssé Theory and Topological Dynamics. In particular it states that the automorphisms group of the Fraïssé limit of a countable Fraïssé family $\mathcal F$ consisting of finite rigid structures is extremely amenable if and only if $\mathcal F$ has some Ramsey property.
Hide abstract.
The previous result has been extended to the dual context of projective Fraïssé Theory by D. Bartošová and A. Kwiatkowska. In such a context we present our work on projective Fraïssé limits of partial orders and their quotients. This is joint work with R. Camerlo. 
April 4th, 2017, 10.3012.00 (Palazzo Campana, Aula Seminari)
G. Basso (Lausanne and Torino) "Projective Fraïssé Limits of Partial Orders", part 2.
The KechrisPestovTodorčević correspondence links Ramsey Theory, Fraïssé Theory and Topological Dynamics. In particular it states that the automorphisms group of the Fraïssé limit of a countable Fraïssé family $\mathcal F$ consisting of finite rigid structures is extremely amenable if and only if $\mathcal F$ has some Ramsey property.
Hide abstract.
The previous result has been extended to the dual context of projective Fraïssé Theory by D. Bartošová and A. Kwiatkowska. In such a context we present our work on projective Fraïssé limits of partial orders and their quotients. This is joint work with R. Camerlo. 
April 11th, 2017, 09.3011.00 (Palazzo Campana, Aula 5)
N. Gambino (Leeds) "An introduction to Voevodsky's univalent type theories".
Around 2006, the Fields Medallist Vladimir Voevodsky introduced a new typetheoretic axiom, called the Univalence Axiom, and formulated an ambitious research programme aimed at developing mathematics within MartinLöf type theories extended with the Univalence Axiom. I will give an overview of these ideas, without assuming any prior knowledge of type theory, focusing on the connections with standard settheoretic foundations. If time allows, I will sketch some recent progress on the attempts to give a relative consistency result for univalent type theories.
Hide abstract. 
April 11th, 2017, 11.3012.30 (Palazzo Campana, Aula 5)
A. Karagila (Hebrew University, Jerusalem) "Models of Bristol".
In a workshop hosted in Bristol back in 2011, the participants outlined a construction of a model of $ZF$ which lies between $L$ and $L[c]$ for a Cohen real $c$, which satisfies that $V \neq L(x)$ for any set $x$. The details of the construction were never fully written. Until now. We will present the key ideas and methods needed to construct the Bristol model, and outline the construction. This will show that the construction can be carried out from ground models far more interesting than $L$ itself.
Hide abstract. 
May 9th, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
M. Di Nasso (Pisa) "Ramsey properties of nonlinear Diophantine equations".
Ramsey theory studies structural combinatorial properties that are preserved under finite partitions. An active area of research in this framework has overlaps with additive number theory, and it focuses on partition properties of the natural numbers related to their semiring structure. We present new results about sufficient and necessary conditions for the partition regularity of Diophantine equations on $\mathbb N$, which extend the classic Rado's Theorem. The goal is to contribute to an overall theory of Ramsey properties of (nonlinear) Diophantine equations that encompasses the known results in this area under a unified framework. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters $\beta \mathbb N$. Necessary conditions are proved by a new technique in nonstandard analysis, based on the relation of $u$equivalence for hypernatural numbers.
Hide abstract. 
May 23rd, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
R. Camerlo (Politecnico di Torino) "Analytic sets and the density function in the Cantor space", part 1.
The density function $ \mathcal D_A$ for measurable subsets $A$ of the Cantor space $2^{ \mathbb N }$ will be presented. It will be shown that the set of all pairs $(K,r)$ with $K$ compact in $2^{ \mathbb N }$ and $r= \mathcal D_K(z)\in (0,1)$ for some $z\in 2^{ \mathbb N }$ is universal for analytic subsets of the real interval $(0,1)$. This is a joint work with A. Andretta.
Hide abstract. 
May 30th, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
R. Camerlo (Politecnico di Torino) "Analytic sets and the density function in the Cantor space", part 2.
The density function $ \mathcal D_A$ for measurable subsets $A$ of the Cantor space $2^{ \mathbb N }$ will be presented. It will be shown that the set of all pairs $(K,r)$ with $K$ compact in $2^{ \mathbb N }$ and $r= \mathcal D_K(z)\in (0,1)$ for some $z\in 2^{ \mathbb N }$ is universal for analytic subsets of the real interval $(0,1)$. This is a joint work with A. Andretta.
Hide abstract. 
June 6th, 2017, 10.3012.00 (Palazzo Campana, Aula 5)
A. Vignati (Paris 7) "Set theoretical dichotomies in $C^*$algebras".
After a brief introduction, we survey the recent progresses in the applications of set theory to the study of automorphisms of corona $C^*$algebras. Corona $C^*$algebras, noncommutative generalizations of the CechStone remainder of a topological space. We show how different set theoretical axioms have an impact on the quantity and the quality of possible automorphisms of such $C^*$algebras, and we infer a serie of dichotomies. This is partly joint work with P. McKenney.
Hide abstract. 
June 26th, 2017, 10.3012.00 (Palazzo Campana, Aula Lagrange)
D. Sinapova (Chicago) "Iterating Prikry forcing".
We will present an abstract approach of iterating Prikry type forcing. Then we will use it to show that it is consistent to have finite simultaneous stationary reflection at $\kappa^+$ with not SCH at $\kappa$. This extends a result of Assaf Sharon. Finally we will discuss how we can bring the construction down to $\aleph_{\omega}$. This is joint work with Assaf Rinot.
Hide abstract. 
July 21st, 2017, 10.0012.00 (Palazzo Campana, Aula S)
Y. Pequignot (UCLA) "$\Sigma^1_2$ sets and countable Borel chromatic numbers".
Analytic sets enjoy a classical representation theorem based on wellfounded relations. I will explain a similar representation theorem for $\Sigma^1_2$ sets due to Marcone. This can be used to answer negatively the primary outstanding question from (Kechris, Solecki and Todorcevic; 1999): the shift graph is not minimal among the graphs of Borel functions which have infinite Borel chromatic number.
Hide abstract. 
November 8th, 2017, 11.3013.00 (Palazzo Campana, Aula 3)
F. Cavallari (TorinoLausanne) "Decidability of regular tree languages in low levels of the Borel and Wadge hierarchy", part 1.
In these seminars I will present all the results that we know about decidability of Borel regular tree languages. We will start with the bottom degrees of the Wadge hierarchy, and we will arrive up to the second level of the Borel hierarchy of the Cantor space.
Hide abstract. 
November 15th, 2017, 11.3013.00 (Palazzo Campana, Aula 3)
F. Cavallari (TorinoLausanne) "Decidability of regular tree languages in low levels of the Borel and Wadge hierarchy", part 2.
In these seminars I will present all the results that we know about decidability of Borel regular tree languages. We will start with the bottom degrees of the Wadge hierarchy, and we will arrive up to the second level of the Borel hierarchy of the Cantor space.
Hide abstract. 
November 29th, 2017, 11.3013.00 (Palazzo Campana, Aula 3)
F. Cavallari (TorinoLausanne) "Decidability of regular tree languages in low levels of the Borel and Wadge hierarchy", part 3.
In these seminars I will present all the results that we know about decidability of Borel regular tree languages. We will start with the bottom degrees of the Wadge hierarchy, and we will arrive up to the second level of the Borel hierarchy of the Cantor space.
Hide abstract. 
December 6th, 2017, 14.3016.00 (Palazzo Campana, Aula 1)
F. Cavallari (TorinoLausanne) "Decidability of regular tree languages in low levels of the Borel and Wadge hierarchy", part 4.
In these seminars I will present all the results that we know about decidability of Borel regular tree languages. We will start with the bottom degrees of the Wadge hierarchy, and we will arrive up to the second level of the Borel hierarchy of the Cantor space.
Hide abstract.
Hide list 2017.

January 8th, 2016, 10.3012.00 (Palazzo Campana, Aula 3)
A. Vignati (York University, Toronto) "C*algebras, forcing axioms and stability".
After a brief introduction we explore the connections between forcing axioms and the study of the group of automorphisms of some particular C*algebras. We connect all of this together with some results in stability theory. This is joint work with Paul McKenney.
Hide abstract. 
March 3rd, 2016, 09.3010.30 (Palazzo Campana, Aula S)
J. Hamkins (City University of New York) "Open determinacy for games on the ordinals".
The principle of open determinacy for class games  twoplayer games of perfect information with plays of length $omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals  is not provable in ZermeloFraenkel set theory ZFC or GödelBernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for firstorder truth, and indeed a transfinite tower of truth predicates for iterated truthabouttruth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over wellfounded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC + $Pi^1_1$comprehension, a proper fragment of KelleyMorse set theory KM. This is joint work with Victoria Gitman. Discussion and commentary can be made there.
Hide abstract. 
March 3rd, 2016, 11.0012.00 (Palazzo Campana, Aula S)
S. Cox (Virginia Commonwealth University) "Layered posets, weak compactness, and Kunen's universal collapse".
A poset \(\mathbb{Q}\) is called \(\kappa\)stationarily layered if the set of regular suborders of $\mathbb{Q}$ is stationary in $P_\kappa(\mathbb{Q})$. Stationary layering implies the Knaster property, and that small sets in the forcing extension are captured by small regular suborders. Layered posets have recently been used to provide a new characterization of weak compactness (CoxLücke 2015), and to prove that any Kunenstyle universal iteration of $\kappa$cc posets  possibly each of size $\kappa$  is $\kappa$cc, provided that $\kappa$ is weakly compact and direct limits are used sufficiently often.(Cox 2015)
Hide abstract. 
April 6th, 2016, 10.3012.30 (Palazzo Campana, Aula 2)
M. Viale (Torino) "Generic absoluteness and boolean names for elements of a Polish space", part 1.
It is common knowledge in the set theory community that there exists a duality relating the commutative C*algebras with the family of Bnames for complex numbers in a boolean valued model for set theory $V^B$. Several aspects of this correlation have been considered in works of the late seventies and early eighties, for example by Takeuti and Jech. Generalizing Jech's results, we extend this duality so to be able to describe the family of boolean names for elements of any given Polish space $Y$ (such as the complex numbers) in a boolean valued model for set theory $V^B$ as a space $C^+(X,Y)$ consisting of functions $f$ whose domain $X$ is the Stone space of $B$, and whose range is contained in $Y$ modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of $C^+(X,Y)$.
Hide abstract. 
April 13th, 2016, 10.3012.30 (Palazzo Campana, Aula 2)
M. Viale (Torino) "Generic absoluteness and boolean names for elements of a Polish space", part 2.
It is common knowledge in the set theory community that there exists a duality relating the commutative C*algebras with the family of Bnames for complex numbers in a boolean valued model for set theory $V^B$. Several aspects of this correlation have been considered in works of the late seventies and early eighties, for example by Takeuti and Jech. Generalizing Jech's results, we extend this duality so to be able to describe the family of boolean names for elements of any given Polish space $Y$ (such as the complex numbers) in a boolean valued model for set theory $V^B$ as a space $C^+(X,Y)$ consisting of functions $f$ whose domain $X$ is the Stone space of $B$, and whose range is contained in $Y$ modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of $C^+(X,Y)$.
Hide abstract. January 8th, 2016, 9.3010.30 (Palazzo Campana, Aula 3)
A. Vignati (York University, Toronto), "C*algebras, forcing axioms and stability".
After a brief introduction we explore the connections between forcing axioms and the study of the group of automorphisms of some particular C*algebras. We connect all of this together with some results in stability theory. This is joint work with Paul McKenney.
Hide abstract.March 3rd, 2016, 9.3010.30 (Palazzo Campana, Aula S)
Joel David Hamkins (City University of New York), "Open determinacy for games on the ordinals".
The principle of open determinacy for class games  twoplayer games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals  is not provable in ZermeloFraenkel set theory ZFC or GödelBernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for firstorder truth, and indeed a transfinite tower of truth predicates for iterated truthabouttruth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over wellfounded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC + $\Pi^1_1$comprehension, a proper fragment of KelleyMorse set theory KM.
Hide abstract.
This is joint work with Victoria Gitman. Discussion and commentary can be made there.March 3rd, 2016, 11.0012.00 (Palazzo Campana, Aula S)
Sean Cox(Virginia Commonwealth University), "Layered posets, weak compactness, and Kunen's universal collapse".
A poset $\mathbb{Q}$ is called $\kappa$stationarily layered if the set of regular suborders of $\mathbb{Q}$ is stationary in $P_\kappa(\mathbb{Q})$. Stationary layering implies the Knaster property, and that small sets in the forcing extension are captured by small regular suborders. Layered posets have recently been used to provide a new characterization of weak compactness (CoxLücke 2015), and to prove that any Kunenstyle universal iteration of $\kappa$cc posets  possibly each of size $\kappa$  is $\kappa$cc, provided that $\kappa$ is weakly compact and direct limits are used sufficiently often.(Cox 2015)
Hide abstract.April 6th, 2016, 10.3012.30 (Palazzo Campana, Aula 2)
M. Viale (Torino), "Generic absoluteness and boolean names for elements of a Polish space", part 1.
It is common knowledge in the set theory community that there exists a duality relating the commutative C*algebras with the family of Bnames for complex numbers in a boolean valued model for set theory V^B. Several aspects of this correlation have been considered in works of the late seventies and early eighties, for example by Takeuti and Jech. Generalizing Jech's results, we extend this duality so to be able to describe the family of boolean names for elements of any given Polish space Y (such as the complex numbers) in a boolean valued model for set theory V^B as a space C^+(X,Y) consisting of functions f whose domain X is the Stone space of B, and whose range is contained in Y modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of C^+(X,Y).
Hide abstract.April 13th, 2016, 10.3012.30 (Palazzo Campana, Aula 2)
M. Viale (Torino), "Generic absoluteness and boolean names for elements of a Polish space", part 2.
April 20th, 2016, 10.3012.30 (Palazzo Campana, Aula S)
B. Velickovic (Paris 7  Denis Diderot), "Precipitousness of the nonstationary ideal".
May 11th, 2016, 10.3012.30 (Palazzo Campana, Aula 2)
M. Viale (Torino), "Forcing the truth of a weak form of Schanuel's conjecture", part 1.
May 13th, 2016, 10.0012.00 (Palazzo Campana, Auletta seminario geometria)
S. Steila (Bern), "An introduction to Operational Set Theory".
We will present some operational set theories (introduced by Feferman) and compare their strength with classical set theories.
Hide abstract.May 18th, 2016, 10.3012.30 (Palazzo Campana, Aula 2)
M. Viale (Torino), "Forcing the truth of a weak form of Schanuel's conjecture", part 2.
Schanuel's conjecture states that the transcendence degree over the rationals Q of the 2ntuple (a_1;...; a_n; exp(a_1) ; ; exp(a_n)) is at least n for all a_1;...; a_n in the complex numbers C which are linearly independent over Q; if true it would settle a great number of elementary open problems in number theory, among which the transcendence of e over \pi. Wilkie and Kirby have proved that there exists a smallest countable algebraically and exponentially closed subfield K of the complex numbers C such that Schanuel's conjecture holds relative to K (i.e. modulo the trivial counterexam ples, Q can be replaced by K in the statement of Schanuel's conjecture). We prove a slightly weaker result (i.e. that there exists such a countable eld K without specifying that there is a smallest such) using the forcing method and Shoenfield's absoluteness theorem. This result suggests that forcing can be a useful tool to prove theorems (rather than independence results) and to tackle problems in domains which are appar ently quite far apart from set theory.
Hide abstract.
May 24th, 2016, 12.3014.30 (Palazzo Campana Aula 3)
J. Gismatullin (Wroclaw), "On the notion of metric ultraproduct".
I am going to explain the notion of metric ultraproduct of structures (mainly groups) and give applications.

May 25th, 2016, 10.3012.30 (Palazzo Campana Aula 1)
J. Gismatullin (Wroclaw), "Approximation properties of groups".
I will present some recent results on groups with good metric approximation properties, called (weak) sofic and (weak) hyperlinear groups. The notion of a sofic group was introduced by B. Weiss and M. Gromov, in the connection with the problem posed by W. Gottschalk on Bernoulli shifts. Recently several conjectures from group theory and topological dynamics have been solved for sofic groups. I will explain modeltheoretic approach to problems around this topic.
May 26th, 2016, 16.0017.00 (Palazzo Campana, Aula Magna)
S. Thomas (Rutgers), "A descriptive view of infinite dimensional group representations".
If G is a finite group, then G has finitely many inequivalent irreducible representations as a group of matrices over a finite dimensional complex vector space, and each finite dimensional representation of G can be expressed uniquely as a direct sum of finitely many irreducible representations. Unfortunately, many infinite groups have no nontrivial finite dimensional representations and so it is necessary to consider their infinite dimensional representations. However, the basic theory of the infinite dimensional representations of infinite groups is much less satisfactory. In particular, such a group typically has uncountably many irreducible infinite dimensional representations. In this talk, I will consider questions such as:
Hide abstract.
(i) For which infinite groups G is it possible to classify its irreducible representations?
(ii) What does it mean to classify an uncountable set of irreducible representations?
Along the way, we will see the representation theorists Mackey, Glimm and Effros making fundamental contributions to descriptive set theory, and the descriptive set theorists Kechris and Hjorth making fundamental contributions to representation theory.
This talk will be aimed at a general mathematical audience. In particular, I will not assume a prior knowledge of either representation theory or descriptive set theory.June 1st, 2016, 15.0016.00 (Palazzo Campana, Sala S)
S. Thomas (Rutgers), "The isomorphism and biembeddability relations for finitely generated groups".
I will discuss the isomorphism and biembeddability relations for various classes of finitely generated groups. In particular, I will point out a recursiontheoretic obstacle to proving that the isomorphism relation for finitely generated simple groups is complicated.
Hide abstract.June 7th, 2016, 10.3012.30 (Palazzo Campana, Aula 3)
R.Carroy (Torino), "Strongly surjective linear orders", part 1.
When a linear order has an increasing surjection onto each of its suborders we say that it is strongly surjective. We prove that countable strongly surjective orders are the union of an analytic and a coanalytic set, and that moreover they are complete for this class of sets.
Hide abstract.
We also prove under PFA the existence of uncountable strongly surjective orders.June 8th, 2016, 10.3012.30 (Palazzo Campana, Aula 3)
R.Carroy (Torino), "Strongly surjective linear orders", part 2.
June 22nd, 2016, 10.3012.30 (Palazzo Campana, Aula 1)
R.Carroy (Torino), "Strongly surjective linear orders", part 3.
June 27th, 2016, 10.3012.30 (Palazzo Campana, Aula S)
J.Bagaria (Barcelona), "Structural Reflection and remarkable cardinals".
I will present the principle of Structural Reflection (SR) as a natural general framework for the study of large cardinal principles. In particular, I will focus on some recent work, done in collaboration with Victoria Gitman (New York) and Ralf Schindler (Muenster) on the characterization of remarkable cardinals in terms of SR.
Hide abstract.June 27th, 2016, 14.3016.30 (Palazzo Campana, Aula S)
R.Carroy (Torino), "Strongly surjective linear orders", part 4.
July 26th, 2016, 11.0012.30 (Palazzo Campana, Aula S)
M. Lupini (Caltech), "The omitting types theorem and the entropy realization problem".
I will present an application of the omitting types theorem for the logic for metric structures to the Furstenberg entropy realization problem: the set of values attained by the Furstenberg entropy on boundary stationary actions is a closed set. This is joint work with Peter Burton and Omer Tamuz. No in depth knowledge of ergodic theory or the logic for metric structures will be assumed.
Hide abstract.October 19th, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino), "Classes of Polish spaces under effective Borel isomorphism", part 1.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract.
In this first talk we will discuss some basic facts of effective descriptive set theory, and we will explain the motivation for investigating the problem of effective Borel isomorphism. We will also introduce some basic tools and time permitting we will present our first counterexample.October 26th, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino), "Classes of Polish spaces under effective Borel isomorphism", part 2.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract.November 2nd, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino), "Classes of Polish spaces under effective Borel isomorphism", part 3.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract.November 9th, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino), "Classes of Polish spaces under effective Borel isomorphism", part 4.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract.November 16th, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino), "Classes of Polish spaces under effective Borel isomorphism", part 5.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract.November 23rd, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
F. Calderoni (Torino), "On the complexity of the biembeddability between torsionfree abelian groups of uncountable size", part 1.
Working in the framework of Generalized Descriptive Set Theory, we discuss the problem of determining the complexity of the biembeddability between torsionfree abelian groups of uncountable size.
Hide abstract.November 30th, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
V. Gregoriades (Torino), "Classes of Polish spaces under effective Borel isomorphism", part 6.
It is a fundamental fact in descriptive set theory that every uncountable Polish space is Borel isomorphic to the Baire space. As it turns out, the effective (descriptive set theoretic) version of this result is far from being true. In fact the relation induced by effective Borel injections carries a rich structure, and includes infinite decreasing sequences as well as antichains.
Hide abstract.December 7th, 2016, 11.3013.00 (Palazzo Campana, Aula 3)
F. Calderoni (Torino), "On the complexity of the biembeddability between torsionfree abelian groups of uncountable size", part 2.
Working in the framework of Generalized Descriptive Set Theory, we discuss the problem of determining the complexity of the biembeddability between torsionfree abelian groups of uncountable size.
Hide abstract.December 21st, 2016, 11.3013.00 (Palazzo Campana, Aula 5)
F. Calderoni (Torino), "On the complexity of the biembeddability between torsionfree abelian groups of uncountable size", part 3.
Working in the framework of Generalized Descriptive Set Theory, we discuss the problem of determining the complexity of the biembeddability between torsionfree abelian groups of uncountable size.
Hide abstract.
Hide list 2016.
March 2nd 2015 (1113, Aula 1, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 1.
March 6th 2015 (1416, Aula 3, Palazzo Campana)
Andrea Medini (Vienna), "Dropping polishness". See abstract.
Classical descriptive set theory studies the subsets of complexity Gamma of a Polish space X, where Gamma is one of the (boldface) Borel or projective pointclasses. However, the definition of a Gamma subset of X extends in a natural way to spaces X that are separable metrizable, but not necessarily Polish.
Hide abstract.
When one "drops Polishness", many classical results suggest new problems in this context. We will discuss some early examples, then focus on the perfect set property. More precisely, we will determine the status of the statement
"For every separable metrizable X, if every Gamma subset of X has the perfect set property then every Gamma' subset of X has the perfect set property" as Gamma, Gamma' range over all pointclasses of complexity at most analytic or coanalytic.March 9th 2015 (1113, Aula C, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 2.
March 20th 2015 (1113, Aula C, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 3.
March 20th 2015 (1416, Aula 3, Palazzo Campana)
Vincenzo Mantova (Pisa), "Surreal numbers, derivations and transseries". See abstract (in Italian).
I surreali di Conway sono una classe ''No'' di numeri originariamente pensati come configurazioni di un gioco, ma dotati di una struttura naturale di campo ordinato e di una funzione esponenziale che li rendono un modello mostro della teoria di (R,exp). Vari autori hanno congetturato che No puo' essere descritto come campo di transserie e ci sia una struttura di campo differenziale simile a quella dei campi di Hardy. In un lavoro in collaborazione con Alessandro Berarducci risolviamo entrambi i problemi e dimostriamo anche che la derivazione naturale e' Liouvillechiusa, ovvero surgettiva.
Hide abstract.March 23rd 2015 (1113, Aula C, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 4.
March 26th 2015 (Thursday!, 1416, Aula 1, Palazzo Campana)
Artem Chernikov (Paris), "Action of the automorphism group of a countable omegacategorical structure on its space of types". See abstract.
We discuss topological dynamics of the action of the automorphism group of a countable omegacategorical structure on its space of types. In particular, we consider a definable counterpart of the KechrisPestovTodorcevic correspondence and the effect of various modeltheoretic assumptions (stability, NIP, etc).
Hide abstract.April 10th 2015 (1113, Sala S, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 5.
April 10th 2015 (1416, Aula 3, Palazzo Campana)
Ludomir Newelski (Wroclaw), "Model theory and topological dynamics". See abstract.
Among the central notions of (stable) model theory are these of (Morley) ranks, forking (independence), generic types/sets in stable groups. These work well in the stable case, but not so anymore in general.
Hide abstract.
Here topological dynamics comes to the rescue. In particular, in the case of definable groups, the definable topological dynamics provides the correct counterparts/generalizations of the notion of generic types/sets. The notions of topological dynamics are related to modeltheoretic properties of groups. Also, the modeltheoretic setup suggests new ideas in topological dynamics itself. I will discuss the notion of a strongly generic set and refer it to existence of bounded orbits and definable amenability of a group.April 13th 2015 (1113, Aula C, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 6.
April 17th 2015 (1113, Sala S, Palazzo Campana)
Vassilis Gregoriades (Darmstadt), "Effective descriptive set theory: aspects of the past and directions for the future". See abstract.
In this seminar talk we will present the basic facts of effective descriptive set theory, explain the main differences from the classical theory, and review some of its cornerstone results. We will also present some recent developments of the effective theory as well as some new applications. Finally we will discuss prospects for future research.
Hide abstract.April 20th 2015 (1113, Aula C, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 7.
April 24th 2015 (1113, Sala S, Palazzo Campana)
Yann Pequignot (Lausanne/Paris), "A Wadge hierarchy for second countable spaces". See abstract.
We define a notion of reducibility for subsets of a second countable T_{0} topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the HausdorffKuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces.
Hide abstract.
However in virtually every second countable T_{0} space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line or the Scott Domain.April 24th 2015 (1416, Aula 3, Palazzo Campana)
Kevin Fournier (Lausanne/Paris), "Wadge Hierarchy of differences of coanalytic sets". See abstract.
We begin the fine analysis of non Borel pointclasses. Working under coanalytic determinacy, we describe the Wadge hierarchy of the class of increasing differences of coanalytic subsets of the Baire space by extending results obtained by Louveau for the Borel sets.
Hide abstract.April 27th 2015 (1113, Aula 1, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 8.
May 4th 2015 (1113, Aula 1, Palazzo Campana)
L. Motto Ros, "On isometry and isometric embeddability between metric and ultrametric Polish spaces", part 9.
May 8th 2015 (1416, Aula 3, Palazzo Campana)
Philipp Lücke (Bonn), "Fragments of the Forcing Theorem for Class Forcings". See abstract.
Class forcing generalizes set forcing by allowing partial orders that are proper classes and requiring generic filters to intersect all dense subclasses of these partial orders. hile it is easy to see that such forcings need not preserve the axioms of ZFC, the question whether certain fragments of the forcing theorem hold for all class forcings was open. I will present results that answer this question by showing that all aspects of the forcing theorem can fail for class forcings. More specifically, there is a class forcing whose forcing relation is not definable and there is a class forcing that does not satisfy the truth lemma. Moreover, I will show that the validity of the forcing theorem for a given class forcing is equivalent to the existence of definable boolean completion of that forcing. This is joint work with Peter Holy, Regula Krapf, Ana Njegomir and Philipp Schlicht (Bonn).
Hide abstract.May 15th 2015 (1113, Aula 3, Palazzo Campana)
Heike Mildenberger (Freiburg), "Subforcings of BlassShelah forcing". See abstract.
The second components in BlassShelah forcing, the socalled pure parts of the conditions, are normed sequences of finite sets of finite sets of natural numbers. (The doubling is not a mistake.) We consider centred subcollections ${\mathcal C}$, so that BlassShelah forcing with pure parts taken from ${\mathcal C}$ preserves certain $P$points and can be used to build up, along a countable support iteration, a large ultrafilter.
Hide abstract.May 21st 2015 (Thursday!, 1416, Aula 1, Palazzo Campana)
Julien Melleray (Lyon), "Full groups of minimal homeomorphisms and descriptive set theory". See abstract.
A homeomorphism of a Cantor space is said to be minimal if all of its orbits are dense. Trying to understand the corresponding equivalence relation (given by the orbit partition) gives rise to interesting problems, and in particular leads one to consider the "full group" of this relation, that is, the group of all homeomorphisms which map each orbit onto itself. I will discuss some descriptivesettheoretic properties of this group (it is coanalytic non Borel, and does not admit a compatible Polish topology), and try to explain why its closure inside the homeomorphism group of the ambient Cantor space is interesting to study. If time permits, I will discuss some questions related to the Borel complexity of certain natural equivalence relations (namely, isomorphism and orbit equivalence of minimal homeomorphisms). A large part of the talk will be based on joint work with T. Ibarlucia (Lyon); no prerequisites about topological dynamics will be assumed, and I wil try to avoid getting into technical descriptivesettheoretic discussions.
Hide abstract.May 27th 2015 (Wednesday!, 1113, Aula 2, Palazzo Campana)
David Asperó (East Anglia), "Some uses of homogeneous forcing". See abstract.
I will present some applications of homogeneous forcing notions in one or two contexts: The context of $\Omega$complete theories for $H(\omega_2)$ (to what extent must these theories be unique?) and, possibly, the context of relative definability (if a is some object satisfying some given property P(x), can I define an object satisfying another given property Q(x) from a?).
Hide abstract.May 29th 2015 (1113, Aula 3, Palazzo Campana)
Pierre Simon (Lyon), "Order and stability in NIP theories". See abstract.
The class of NIP theories was defined by Shelah in the 70s, but has stayed in the background for some 30 years. In the last decade, it has received a lot of attention from model theorists fueled in particular by the growing interest in valued fields. This class of theories contains both stable and ominimal theories. The intuition driving its study is that the properties of NIP structures should somehow be a combination of stability and ominimality. I will survey results obtained in the last 5 years that give evidence towards this idea and in fact try to make it precise by decomposing types into stable and orderlike components.
Hide abstract.June 5th 2015 (1416, Aula 3, Palazzo Campana)
Lionel Nguyen van Thè (Marseille), "Ramseytype phenomena from fixed points in compactifications". See abstract.
Ramsey theory (which is, roughly, the study of the necessary appearance of very organized substructures inside of any sufficiently large structure) has lately largely benefited from its connection to various other fields, especially dynamics and functional analysis. In this talk, I will illustrate this further by showing how the existence of fixed points in certain group compactifications allows to derive new Ramseytype results.
Hide abstract.June 17th 2015 (1113, Aula 3, Palazzo Campana)
Guido Gherardi (Universität der Bundeswehr München), "Betting sometimes may help: on infinite certainties and conscious mistakes". See abstract.
Mathematical statements of type "For all x in X there exists a y in Y" defines settheoretically functions in an obvious way. What is less trivial is the investigation about their effective level of computability. To this goal I am going to show different kinds of Turing machines aims at computing functions determined by classical theorems from analysis and topology. As a particular case study I will focus on Las Vegas computability, by extending to the continuum the well known corresponding notion usually used for discrete computations. As an application example, I am going to analyze the classical Vitali's Theorem "every Vitali's covering of a Lebesguemeasurable set of real numbers contains a subsequence of open disjoint members that covers the given set up to measure zero". Joint work with V. Brattka e R. Hölzl.
Hide abstract.June 19th 2015 (1113, Aula 3, Palazzo Campana)
Kostas Tsaprounis (Salvador de Bahia), "On ultrahuge cardinals". See abstract.
Starting with the wellknown notion of a superhuge cardinal, we strengthen it by requiring that the witnessing elementary embeddings are, in addition, sufficiently superstrong above their target j(k). This modification leads us to a new large cardinal which we call ultrahuge. In this talk, we introduce the notion of ultrahugeness and study its placement in the usual large cardinal hierarchy, while also show that some standard techniques apply nicely in its context as well. Moreover, we further look at the corresponding C^(n)versions of ultrahugeness; as it turns out, these constitute a (proper) refinement of the large cardinal hierarchy between the notions of almost 2hugeness and superhugeness. Hide abstract.
23 giugno 2015 (1113, Aula Magna, Palazzo Campana)
Andrés Villaveces (Bogotá), "Categoricity, between model theory and set theory". See abstract.
The work toward the Categoricity Conjecture for Abstract Elementary Classes is entangled with both large cardinals and forcing. A family of connections to large cardinal properties has been started by Boney around locality notions such as tameness and type  shortness. The behavior of these locality notions is akin (but not equivalent) to tree properties and reflection principles. The second kind of connections (to forcing) arises in two ways at least: from strong forms of collapse preserving tameness and from applications of forcing axioms for categoricity at small cardinals. This last part is very much work in progress. Hide abstract.
25 giugno 2015 (1012, Aula 2, Palazzo Campana)
Krzysztof Krupinski (Wroclaw), "Topological dynamics and Borel cardinalities in model theory." Abstract.
Newelski introduced methods and ideas from topological dynamics to the context of definable groups. I will recall some fundamental issues concerning this approach, and I will present a few deeper results from my joined paper with Anand Pillay written last year, which relate the so called generalized Bohr compactification of the given definable group to its modeltheoretic connected components. Then I will discuss more recent (analogous) results for the group of automorphisms of the monster model, relating notions from topological dynamics to various Galois groups of the theory in question. As an application, I will present a general theorem concerning Borel cardinalities of Borel, bounded equivalence relations, which gives answers to some questions of Kaplan and Miller and of Rzepecki and myself. This theorem was not accessible by the methods used so far in the study of Borel cardinalities of Borel, bounded equivalence relations (by Kaplan, Miller, Pillay, Simon, Solecki, Rzepecki and myself). The topological dynamics for the group of automorphisms of the monster model and its applications to Borel cardinalities are planned to be contained in my future joint paper with Anand Pillay and Tomasz Rzepecki. Hide abstract.
November 11th, 2015 (15.30, Aula Buzano, DISMA, Politecnico)
Vincenzo Dimonte (Vienna), "Large cardinals in mathematics and in infinite combinatorics". Slides
November 13th, 2015 (10.00, Aula Buzano, DISMA, Politecnico)
Vincenzo Dimonte (Vienna), "Very large cardinals and forcing".
Hide list 2015.
Student seminar
Upcoming:
Past:
January 8th, 2016, 10.3012.30 (Palazzo Campana, Aula 3)
F. Cavallari, "Alternating automata and weak alternating automata".
January 15th, 2016, 10.3012.30 (Palazzo Campana, Aula 3)
F. Cavallari, "Alternating automata and weak alternating automata", part 2.
January 29th, 2016, 10.3012.30 (Palazzo Campana, Aula 5)
G. Audrito, "Filter systems: towers, extenders and so on.", part 1.
February 12th, 2016, 10.3012.30 (Palazzo Campana, Aula 5)
G. Audrito, "Filter systems: towers, extenders and so on.", part 2.
February 19th, 2016, 10.3012.30 (Palazzo Campana, Aula 5)
G. Audrito, "Filter systems: towers, extenders and so on.", part 3.
February 26th, 2016, 10.3012.30 (Palazzo Campana, Aula 5)
G. Audrito, "Filter systems: towers, extenders and so on.", part 4.
March 4th, 2016, 14.3016.30 (Palazzo Campana)
F. Calderoni, "How difficult is it to classify unitary irreducible representation?", part 1.
March 11th, 2016, 14.3016.30 (Palazzo Campana)
F. Calderoni, "How difficult is it to classify unitary irreducible representation?", part 2.
March 18th, 2016, 14.3016.30 (Palazzo Campana)
F. Calderoni, "How difficult is it to classify unitary irreducible representation?", part 3.
March 22nd, 2016, 1416 (Palazzo Campana, Aula 4)
F. Calderoni, "How difficult is it to classify unitary irreducible representation?", part 4.
Hide list 2016.
January 12th 2015 (Palazzo Campana)
S. Steila, "Definable versions of CHequivalences", part 1.
January 14th 2015 (Palazzo Campana)
S. Steila, "Definable versions of CHequivalences", part 2.
January 16th 2015 (Palazzo Campana)
S. Steila, "Definable versions of CHequivalences", part 3.
January 21st 2015 (Palazzo Campana)
G. Carotenuto, "Density sets of reals", part 1.
January 26th 2015 (Palazzo Campana)
G. Carotenuto, "Density sets of reals", part 2.
January 30th 2015 (Palazzo Campana)
G. Carotenuto, "Density sets of reals", part 3.
March 6th 2015 (1113, Sala S, Palazzo Campana)
G. Audrito, "Resurrection axioms and generic absoluteness results", part 1.
March 13th 2015 (1113, Sala S, Palazzo Campana)
G. Audrito, "Resurrection axioms and generic absoluteness results", part 2.
March 16th 2015 (1113, Aula 3, Palazzo Campana)
G. Audrito, "Resurrection axioms and generic absoluteness results", part 3.
March 27th 2015 (1113, Aula C, Palazzo Campana)
G. Audrito, "Resurrection axioms and generic absoluteness results", part 4.
March 30th 2015 (1113, Aula C, Palazzo Campana)
G. Audrito, "Resurrection axioms and generic absoluteness results", part 5.
May 11th 2015 (1113, Aula 1, Palazzo Campana)
F. Calderoni, "Complete analytic quasiorders", part 1.
May 18th 2015 (1113, Aula 1, Palazzo Campana)
F. Calderoni, "Complete analytic quasiorders", part 2.
May 20th 2015 (1113, Aula 2, Palazzo Campana)
F. Calderoni, "Complete analytic quasiorders", part 3.
May 25th 2015 (1113, Aula 1, Palazzo Campana)
F. Calderoni, "Complete analytic quasiorders", part 4.
May 29th, 2015 (14.3016.30, Aula 3, Palazzo Campana)
Hugo Nobrega (Amsterdam), "Game characterizations of functions of finite Baire class". See abstract.
Game characterizations of classes of functions in Baire space have an established tradition in descriptive set theory, especially through the work of Wadge, Duparc, Andretta, and Motto Ros, among others. In his PhD thesis, Semmes introduced the tree game which characterizes the Borel functions, and a certain restriction of this game which characterizes the Baire class 2 functions.
Hide abstract.
In this talk, we show how to restrict the tree game in order to characterize the Baire class n functions, for each finite n. This is done in a uniform way with the help of a certain operation on trees, called the pruning derivative, which we introduce.
We would like to acknowledge that similar results have independently been proved by Louveau and Semmes.December 1st, 2015, 8.3010.30 (Palazzo Campana, Aula 2)
F. Cavallari, "Introduction to automata on infinite words".
December 15th, 2015, 8.3010.30 (Palazzo Campana, Aula 2)
F. Cavallari, "Introduction to automata on infinite trees".
2014
December 1st 2014 (Palazzo Campana)
F. Calderoni, "Liberable groups", part 1.
December 5th 2014 (Palazzo Campana)
F. Calderoni, "Liberable groups", part 2.
December 12th 2014 (Palazzo Campana)
F. Calderoni, "Liberable groups", part 3.
Hide list 2014.