TBA

October 11th, 2019, 14.30-16.30 (Palazzo Campana, Aula S)

Abstract.**R. Mennuni (Leeds)**"Product of invariant types modulo domination-equivalence".In the same way that types generalise ultrafilters, the product of invariant types and domination equivalence generalise, respectively, the tensor product of ultrafilters and Rudin-Keisler equivalence. In stable theories, the latter is a congruence with respect to the former and quotienting yields a well-defined, commutative semigroup.

The interest in this quotient in the unstable case comes from [1], where it was computed in the (unstable) theory of algebraically closed valued fields, obtaining a decomposition result in terms of residue field and the value group. In this talk, I will present the main results from [2]: a counterexample showing that the quotient semigroup need not be well-defined in general, and a first development of the general theory of how this product and this equivalence relation interact.

[1] D. Haskell, E. Hrushovski and D. Macpherson. Stable Domination and Independence in Algebraically Closed Valued Fields, volume 30 of Lecture Notes in Logic. Cambridge University Press (2008).

[2] R. Mennuni. Product of invariant types modulo domination- equivalence. To appear in Archive for Mathematical Logic. Available online hereOctober 4th, 2019, 14.30-16.30 (Palazzo Campana, Aula S)

Abstract.**L. Luperi Baglini (Milan)**"Nonstandard combinatorics ".Results in Ramsey theory and combinatorics can be studied using techniques coming from several different areas. Since Jin's work on piecewise syndetic sets, the use of nonstandard methods became one of the most useful approaches in this area. In this talk, I will give a brief presentation of the most important ideas behind this approach, focusing in particular on applications to the so-called "partition regularity" of equations.

June 25th, 2019, 10.30-12.30 (Palazzo Campana, Aula Spallanzani)

Abstract.**A. Chernikov (UCLA)**"N-dependent theories".A first-order theory is n-dependent if the edge relation of an infinite random n-hypergraph is not definable in any of its models. N-dependence is a strict hierarchy increasing with n, with the first level corresponding to the well-studied class of NIP theories. I will give a survey of recent work on n-dependent theories establishing connections to higher-arity generalizations of VC-dimension and hypergraph regularity in combinatorics (joint with Henry Towsner) and on understanding which algebraic structures are n-dependent (joint with Nadja Hempel).

May 31st, 2019, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**G. Paolini (Turin)**"First-Order Aspects of Coxeter Group ", part 3.We present some recent results joint with S. Shelah and B. Muehlherr on the first-order model theory of Coxeter groups. In particular, we characterize the superstable Coxeter groups of finite rank, and investigate questions of homogeneity and existence of prime models in right-angled Coxeter groups.

May 28th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**V. Marra (Milan)**"Stone duality from the point of view of mathematical logic", part 2.Stone duality, established by Marshall Stone in 1937, asserts that the category of Boolean algebras and their homomorphisms is dually equivalent to the category of Stone spaces (=compact Hausdorff zero-dimensional spaces) and their continuous maps. Historically, Stone duality is a result of paramount importance in the mathematics of the 20th century: a host of other duality theorems in various fields were proved following Stone’s treatment. In this talk I give a leisurely introduction to Stone duality, emphasising its conceptual meaning in mathematical logic.

May 27th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**V. Marra (Milan)**"Stone duality from the point of view of mathematical logic", part 1.Stone duality, established by Marshall Stone in 1937, asserts that the category of Boolean algebras and their homomorphisms is dually equivalent to the category of Stone spaces (=compact Hausdorff zero-dimensional spaces) and their continuous maps. Historically, Stone duality is a result of paramount importance in the mathematics of the 20th century: a host of other duality theorems in various fields were proved following Stone’s treatment. In this talk I give a leisurely introduction to Stone duality, emphasising its conceptual meaning in mathematical logic.

May 24th, 2019, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**G. Paolini (Turin)**"First-Order Aspects of Coxeter Group ", part 2.We present some recent results joint with S. Shelah and B. Muehlherr on the first-order model theory of Coxeter groups. In particular, we characterize the superstable Coxeter groups of finite rank, and investigate questions of homogeneity and existence of prime models in right-angled Coxeter groups.

May 24th, 2019, 10.30-12.30 (Palazzo Campana, Aula 2)

Abstract.**G. Rosolini (Genoa)**"Categorie per la logica" (Slides).Il seminario sarà organizzato in due parti da 45 minuti.

Nella prima parte, presento come le iperdottrine permettono di strutturare le nozioni elementari della logica matematica (teorie del prim'ordine, modelli e possibili estensioni).

Nella seconda parte, affronto la presentazione dei modelli di ZF attraverso particolari iperdottrine note con il nome di tripos.May 21st, 2019, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**G. Paolini (Turin)**"First-Order Aspects of Coxeter Group ", part 1.We present some recent results joint with S. Shelah and B. Muehlherr on the first-order model theory of Coxeter groups. In particular, we characterize the superstable Coxeter groups of finite rank, and investigate questions of homogeneity and existence of prime models in right-angled Coxeter groups.

April 16th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**R. Carroy (Vienna)**"The Open Graph Dichotomy and the second level of the Borel hierarchy".I will sketch a game-theoretical proof of the open graph dichotomy for box-open hypergraphs, and explain how it can be used to obtain various descriptive-set-theoretical dichotomies at the second level of the Borel hierarchy. This shows how to generalize these dichotomies from analytic metric spaces to separable metric spaces by working under the axiom of determinacy. If time allows it, I will also discuss some connections between cardinal invariants and the chromatic number of the graphs at stake. This is joint work with Benjamin D. Miller and Daniel T. Soukup.

March 15th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**T. Tsankov (Paris 7 - Diderot)**"Bernoulli disjointness".The concept of disjointness of dynamical systems (both topological and measure-theoretic) was introduced by Furstenberg in the 60s and has since then become a fundamental tool in dynamics. In this talk, I will discuss disjointness of topological systems of discrete groups. More precisely, generalizing a theorem of Furstenberg (who proved the result for the group of integers), we show that for any discrete group $G$, the Bernoulli shift $2^G$ is disjoint from any minimal dynamical system. This result, together with techniques of Furstenberg, some tools from the theory of strongly irreducible subshifts, and Baire category methods, allows us to answer several open questions in topological dynamics: we solve the so-called "Ellis problem" for discrete groups and also characterize the underlying topological space for the universal minimal flow of discrete groups. No specific knowledge of topological dynamics will be assumed in the talk. This is joint work with Eli Glasner, Benjamin Weiss, and Andy Zucker.

March 5th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**I. Kaplan (Hebrew University of Jerusalem)**"Minimal expansions of torsion-free abelian groups".Joint work with Eran Alouf and Antongiulio Fornasiero

We give sufficient conditions for when an expansion of a somewhat saturated torsion-free abelian group has no proper reducts adding new unary sets. In particular, we show that for any elementary extension $(M,+,<)$ of $(Z,+,<)$, any stable reduct expanding $(M,+)$ is just $(M,+)$.February 19th, 2019, 14.30-16.30 (Palazzo Campana, Aula 4)

Abstract.**A. Fornasiero (Florence)**"Generic derivations on $o$-minimal structures".Let $\mathbb{K}$ be a field. A derivation on $\mathbb{K}$ is a map $d:\mathbb{K}\to \mathbb{K}$ such that $d(x + y) = d(x) + d(y)$ and $d(xy) = x d(y) + d(x) y$ for every $x, y \in \mathbb{K}$. If $\mathbb{K}$ has additional structure, there is a priory no relationship between the derivation $d$ and the additional structure: for instance, if $\mathbb{K}$ is the real field with the exponentiation map exp, $d($exp$ (a))$ could be anything.

When $\mathbb{K}$ is an $o$-minimal structure expanding a field, we may impose the additional condition that $d$ is "compatible" with the extra structure (in the previous example, we impose for instance that $d($exp$ (a)) = d(a) $exp$(a)$).

We show that the theory of compatible derivations (on a fixed $o$-minimal theory) has a model completion (the theory of generic derivations), and describe some properties of these generic derivations.

December 12th, 2018, 14.30-16.30 (Aula Magna)

Abstract.**N. Lavi (Politecnico di Torino)**"Dependent dreams in finite diagrams", part 3.A lot has been said about tameness and wildness, and Shelah believes that the main challenge is to find interesting dividing lines between them. The indepence property is such, and the fact that in its tame side appear many interesting algebraic object serves as a motivation as well as intuition to consider it an interesting and "right" dividing line. The paper "Dependent dreams: recounting of types" [950] is mainly dedicated to this. In a joint work with Shelah and Kaplan, we prove the results in [950] for finite diagrams rather than first order theories, for models of measurable cardinality greater than a strongly compact cardinal.

December 3rd, 2018, 14.30-16.00 (Aula 3)

Abstract.**F. Calderoni (Università degli Studi di Torino)**"On the difficulty of classifying ordered groups".The theory of Borel reducibility has succeeded in establishing the exact complexity of various classification problems throughout mathematics. In this talk we shall analyze the problems of classifying some classes of countable ordered groups up to isomorphism and bi-embeddability. This is done by forming standard Borel spaces of countable ordered groups, and comparing the isomorphism and the bi-embeddability equivalence relations on those spaces with some well-known benchmarks in the class of analytic equivalence relations. We shall discuss recent results, motivations, and open questions.

November 28th, 2018, 14.30-16.30 (Aula Magna)

Abstract.**N. Lavi (Politecnico di Torino)**"Dependent dreams in finite diagrams", part 2.A lot has been said about tameness and wildness, and Shelah believes that the main challenge is to find interesting dividing lines between them. The indepence property is such, and the fact that in its tame side appear many interesting algebraic object serves as a motivation as well as intuition to consider it an interesting and "right" dividing line. The paper "Dependent dreams: recounting of types" [950] is mainly dedicated to this. In a joint work with Shelah and Kaplan, we prove the results in [950] for finite diagrams rather than first order theories, for models of measurable cardinality greater than a strongly compact cardinal.

November 19th, 2018, 14.30-16.30 (Aula 3)

Abstract.**N. Lavi (Politecnico di Torino)**"Dependent dreams in finite diagrams", part 1.A lot has been said about tameness and wildness, and Shelah believes that the main challenge is to find interesting dividing lines between them. The indepence property is such, and the fact that in its tame side appear many interesting algebraic object serves as a motivation as well as intuition to consider it an interesting and "right" dividing line. The paper "Dependent dreams: recounting of types" [950] is mainly dedicated to this. In a joint work with Shelah and Kaplan, we prove the results in [950] for finite diagrams rather than first order theories, for models of measurable cardinality greater than a strongly compact cardinal.

November 16th, 2018, 14.05-16.30 (Palazzo Campana, Aula A)

Abstract.**V. Kanelloupolos (University of Athens)**"Variations and applications of the Hales-Jewett theorem".We will recall the Hales-Jewett theorem and we will present two of its variations. The first one is connected with the Ramsey theory of trees and the second one with Euclidean Ramsey theory.

June 1st, 2018, 10.30-12.30 (Aula A)

Abstract.**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 norm-complete lattice ordered groups, thereby extending Kakutani duality for norm complete vector lattices.

May 31st, 2018, 14.30-15.30 (Politecnico, Aula 1D)

Abstract.**G. Basso (Lausanne and Turin)**"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 pseudo-arc, an interesting indecomposable continuum, and has since constituted a fertile field of research.

May 30th, 2018, 09.30-11.00 (Aula C)

Abstract.**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.

May 18th, 2018, 10.30-12.30 (Palazzo Campana, Aula Lagrange)

Abstract.**V. Giambrone (Turin)**"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.

April 27th, 2018, 10.30-12.30 (Palazzo Campana, Aula Lagrange)

Abstract.**V. Giambrone (Turin)**"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.

April 20th, 2018, 10.30-12.30 (Palazzo Campana, Aula Lagrange)

Abstract.**R. Treglia (Turin)**"$\lambda$-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 Church-Turing 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.

April 6th, 2018, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**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 well-known 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 stem-linked and show that every stem-linked tree forcing has the density property. This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.

April 5th, 2018, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**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.

March 23rd, 2018, 10.30-12.30 (Palazzo Campana, Aula Lagrange)

Abstract.**R. Treglia (Turin)**"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 so-called $\beta$-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 Church-Rosser theorem, will be enunciated.

February 16th, 2018, 10.30-12.00 (Palazzo Campana, Aula 3)

Abstract.**V. Bard (Turin)**"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 76-77) 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 quasi-orders (such as Turing reducibility) because of a Feldman-Moore-like theorem: countable Borel quasi-orders are exactly those quasi-orders 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 bi-reducibility)?". 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.

February 1st, 2018, 14.00-15.30 (Palazzo Campana, Aula 1)

Abstract.**V. Bard (Turin)**"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 76-77) 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 quasi-orders (such as Turing reducibility) because of a Feldman-Moore-like theorem: countable Borel quasi-orders are exactly those quasi-orders 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 bi-reducibility)?". 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.

January 18th, 2018, 14.00-15.30 (Palazzo Campana, Aula 3)

Abstract.**V. Bard (Turin)**"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 76-77) 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 quasi-orders (such as Turing reducibility) because of a Feldman-Moore-like theorem: countable Borel quasi-orders are exactly those quasi-orders 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 bi-reducibility)?". 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.

December 6th, 2017, 14.30-16.00 (Palazzo Campana, Aula 1)

Abstract.**F. Cavallari (Turin-Lausanne)**"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.

November 29th, 2017, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**F. Cavallari (Turin-Lausanne)**"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.

November 15th, 2017, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**F. Cavallari (Turin-Lausanne)**"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.

November 8th, 2017, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**F. Cavallari (Turin-Lausanne)**"Decidability of regular tree languages in low levels of the Borel and Wadge hierarchy", part 1.July 21st, 2017, 10.00-12.00 (Palazzo Campana, Aula S)

Abstract.**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.

June 26th, 2017, 10.30-12.00 (Palazzo Campana, Aula Lagrange)

Abstract.**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.

June 6th, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**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, non-commutative generalizations of the Cech-Stone 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.

May 30th, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**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.

May 23rd, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**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.

May 9th, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**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.

April 11th, 2017, 11.30-12.30 (Palazzo Campana, Aula 5)

Abstract.**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.

April 11th, 2017, 09.30-11.00 (Palazzo Campana, Aula 5)

Abstract.**N. Gambino (Leeds)**"An introduction to Voevodsky's univalent type theories".Around 2006, the Fields Medallist Vladimir Voevodsky introduced a new type-theoretic axiom, called the Univalence Axiom, and formulated an ambitious research programme aimed at developing mathematics within Martin-Löf 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 set-theoretic foundations. If time allows, I will sketch some recent progress on the attempts to give a relative consistency result for univalent type theories.

April 4th, 2017, 10.30-12.00 (Palazzo Campana, Aula Seminari)

Abstract.**G. Basso (Lausanne and Turino)**"Projective Fraïssé Limits of Partial Orders", part 2.The Kechris-Pestov-Todorčević 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.

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.March 28th, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**G. Basso (Lausanne and Turin)**"Projective Fraïssé Limits of Partial Orders", part 1.The Kechris-Pestov-Todorčević 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.

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.March 21st, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**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 Second-Order (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.

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 17th, 2017, 10.30-12.00 (Palazzo Campana, Aula 2)

Abstract.**F. Cavallari (Turin)**"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.

March 14th, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**V. Dimonte (Udine)**"Rank-into-rank axioms and forcing", part 2.At the beginning of the development of rank-into-rank 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.

March 7th, 2017, 10.30-12.00 (Palazzo Campana, Aula 5)

Abstract.**V. Dimonte (Udine)**"Rank-into-rank axioms and forcing", part 1.At the beginning of the development of rank-into-rank 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.

February 22nd, 2017, 11.30-13.00 (Palazzo Campana, Aula C)

Abstract.**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 0-dimensional spaces X and Y this quasi-order is either analytic complete or a better quasi-order. This is a joint work with Yann Pequignot and Zoltan Vidnyanszky.

February 15th, 2017, 11.30-13.00 (Palazzo Campana, Aula C)

Abstract.**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 0-dimensional spaces X and Y this quasi-order is either analytic complete or a better quasi-order. This is a joint work with Yann Pequignot and Zoltan Vidnyanszky.

February 8th, 2017, 11.30-13.00 (Palazzo Campana, Sala S)

Abstract.**J. Somaglia (Milano e Praga)**"Relations between coarse wedge topology on trees and retractional skeletons".I will introduce the classes of Valdivia and non-commutative Valdivia compacta. After that I will recall the definition and some properties of the Coarse wedge topology on trees. Finally a characterization of non-commutative Valdivia trees will be presented.

February 1st, 2017, 11.30-13.00 (Palazzo Campana, Aula 2)

Abstract.**V. Gregoriades (Turin)**"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.

January 25th, 2017, 11.30-13.00 (Palazzo Campana, Sala S)

Abstract.**V. Gregoriades (Turin)**"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.

January 18th, 2017, 11.30-13.00 (Palazzo Campana, Aula 2)

Abstract.**V. Gregoriades (Turin)**"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.

January 11th, 2017, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"Classes of Polish spaces under effective Borel isomorphism", part 7.

December 21st, 2016, 11.30-13.00 (Palazzo Campana, Aula 5)

Abstract.**F. Calderoni (Turin)**"On the complexity of the bi-embeddability between torsion-free 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 bi-embeddability between torsion-free abelian groups of uncountable size.

December 7th, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**F. Calderoni (Turin)**"On the complexity of the bi-embeddability between torsion-free 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 bi-embeddability between torsion-free abelian groups of uncountable size.

November 30th, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"Classes of Polish spaces under effective Borel isomorphism", part 6.November 23rd, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**F. Calderoni (Turin)**"On the complexity of the bi-embeddability between torsion-free 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 bi-embeddability between torsion-free abelian groups of uncountable size.

November 16th, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"Classes of Polish spaces under effective Borel isomorphism", part 5.November 9th, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"Classes of Polish spaces under effective Borel isomorphism", part 4.November 2nd, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"Classes of Polish spaces under effective Borel isomorphism", part 3.October 26th, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"Classes of Polish spaces under effective Borel isomorphism", part 2.October 19th, 2016, 11.30-13.00 (Palazzo Campana, Aula 3)

Abstract.**V. Gregoriades (Turin)**"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.

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.July 26th, 2016, 11.00-12.30 (Palazzo Campana, Aula S)

Abstract.**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.

June 27th, 2016, 14.30-16.30 (Palazzo Campana, Aula S)

Abstract.**R. Carroy (Turin)**"Strongly surjective linear orders", part 4.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.

We also prove under PFA the existence of uncountable strongly surjective orders.June 27th, 2016, 10.30-12.30 (Palazzo Campana, Aula S)

Abstract.**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.

June 22nd, 2016, 10.30-12.30 (Palazzo Campana, Aula 1)

Abstract.**R. Carroy (Turin)**"Strongly surjective linear orders", part 3.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.

We also prove under PFA the existence of uncountable strongly surjective orders.June 8th, 2016, 10.30-12.30 (Palazzo Campana, Aula 3)

Abstract.**R. Carroy (Turin)**"Strongly surjective linear orders", part 2.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.

We also prove under PFA the existence of uncountable strongly surjective orders.June 7th, 2016, 10.30-12.30 (Palazzo Campana, Aula 3)

Abstract.**R. Carroy (Turin)**"Strongly surjective linear orders", part 1.*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.

We also prove under PFA the existence of uncountable strongly surjective orders.June 1st, 2016, 15.00-16.00 (Palazzo Campana, Aula S)

Abstract.**S. Thomas (Rutgers)**"The isomorphism and bi-embeddability relations for finitely generated groups" (Slides).I will discuss the isomorphism and bi-embeddability relations for various classes of finitely generated groups. In particular, I will point out a recursion-theoretic obstacle to proving that the isomorphism relation for finitely generated simple groups is complicated.

May 26th, 2016, 16.00-17.00 (Palazzo Campana, Aula Magna)

Abstract.**S. Thomas (Rutgers)**"A descriptive view of infinite dimensional group representations" (Slides).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:

(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.May 25th, 2016, 10.30-12.30 (Palazzo Campana, Aula 1)

Abstract.**J. Gismatullin (Wrocłav)**"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 model-theoretic approach to problems around this topic.

May 24th, 2016, 12.30-14.30 (Palazzo Campana, Aula 3)

Abstract.**J. Gismatullin (Wrocłav)**"On the notion of metric ultraproduct".I am going to explain the notion of metric ultraproduct of structures (mainly groups) and give applications.

May 18th, 2016, 10.30-12.30 (Palazzo Campana, Aula 2)

Abstract.**M. Viale (Turin)**"Forcing the truth of a weak form of Schanuel's conjecture", part 2.Schanuel's conjecture states that the transcendence degree over the rationals $\mathbb{Q}$ of the $2n$-tuple $(a_1;...; a_n; 2^{a_1} ;...; 2^{a_n})$ is at least $n$ for all $a_1;...; a_n\in\mathbb{C}$ which are linearly independent over $\mathbb{Q}$; 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 $\mathbb{K}$ of the complex numbers $\mathbb{C}$ such that Schanuel's conjecture holds relative to $\mathbb{K}$ (i.e. modulo the trivial counterexamples, $\mathbb{Q}$ can be replaced by $\mathbb{K}$ in the statement of Schanuel's conjecture). We prove a slightly weaker result (i.e. that there exists such a countable field $\mathbb{K}$ 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 apparently quite far apart from set theory.May 13th, 2016, 10.00-12.00 (Palazzo Campana, Auletta seminario Geometria)

Abstract.**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.

May 11th, 2016, 10.30-12.30 (Palazzo Campana, Aula 2)

Abstract.**M. Viale (Turin)**"Forcing the truth of a weak form of Schanuel's conjecture", part 1.Schanuel's conjecture states that the transcendence degree over the rationals $\mathbb{Q}$ of the $2n$-tuple $(a_1;...; a_n; 2^{a_1} ;...; 2^{a_n})$ is at least $n$ for all $a_1;...; a_n\in\mathbb{C}$ which are linearly independent over $\mathbb{Q}$; 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 $\mathbb{K}$ of the complex numbers $\mathbb{C}$ such that Schanuel's conjecture holds relative to $\mathbb{K}$ (i.e. modulo the trivial counterexamples, $\mathbb{Q}$ can be replaced by $\mathbb{K}$ in the statement of Schanuel's conjecture). We prove a slightly weaker result (i.e. that there exists such a countable field $\mathbb{K}$ 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 apparently quite far apart from set theory.April 20th, 2016, 10.30-12.30 (Palazzo Campana, Aula S)

**B. Velickovic (Paris 7 - Denis Diderot)**"Precipitousness of the non-stationary ideal".April 13th, 2016, 10.30-12.30 (Palazzo Campana, Aula 2)

Abstract.**M. Viale (Turin)**"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 B-names 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 $C^+(X,Y)$.

April 6th, 2016, 10.30-12.30 (Palazzo Campana, Aula 2)

Abstract.**M. Viale (Turin)**"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 B-names 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 $C^+(X,Y)$.

March 3rd, 2016, 11.00-12.00 (Palazzo Campana, Aula S)

Abstract.**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 (Cox-Lücke 2015), and to prove that any Kunen-style*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)March 3rd, 2016, 09.30-10.30 (Palazzo Campana, Aula S)

Abstract.**J. Hamkins (City University of New York)**"Open determinacy for games on the ordinals".The principle of open determinacy for class games - two-player 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 Zermelo-Fraenkel set theory ZFC or Gödel-Bernays 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 first-order truth, and indeed a transfinite tower of truth predicates for iterated truth-about-truth, 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 well-founded 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 Kelley-Morse set theory KM. This is joint work with Victoria Gitman. Discussion and commentary can be made there.

January 8th, 2016, 10.30-12.00 (Palazzo Campana, Aula 3)

Abstract.**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.

November 13th, 2015, 15.30-17.00 (Politecnico, DISMA, Aula Buzano)

Abstract.**V. Dimonte (Vienna)**"I grandi cardinali in matematica e in combinatoria infinita" (Slides).La teoria degli insiemi, ramo della logica matematica, ha fin dalla sua fondazione un duplice ruolo in matematica. Si occupa di un'analisi rigorosa dell'infinito e di tutte le sue derivazioni, ma anche delle fondamenta, costruendo una teoria su cui tutta la matematica possa basarsi senza timore di paradossi. La teoria dei grandi cardinali è in questo momento all'avanguardia in entrambi i ruoli, ed è la principale area di ricerca per la consistenza relativa di proposizioni matematiche, con ricadute su diverse aree come analisi, algebra o topologia. Il seminario sarà un veloce excursus di questa teoria e di alcuni suoi successi, con un accento particolare sui grandi cardinali più potenti ed estremi. Non verrà presupposta nessuna conoscenza di logica matematica.

November 13th, 2015, 10.00-12.00 (Politecnico, DISMA, Aula Buzano)

Abstract.**V. Dimonte (Vienna)**"Cardinali molto grandi e forcing".A continuazione del primo seminario, verranno esposti con più dettagli $I0$, il grande cardinale in vetta alla gerarchia, ed alcuni risultati recenti sulle sue interazioni con il forcing, specialmente il forcing di Easton ed il forcing di Prikry, illustrando anche un metodo generale per avere I1 insieme a diverse proprietà combinatoriche, come Diamond, Square o la Tree property.

June 25th, 2015, 10.00-12.00 (Palazzo Campana, Aula 2)

Abstract.**K. Krupiński (Wrocłav)**"Topological dynamics and Borel cardinalities in model theory".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 model-theoretic 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.

June 23rd, 2015, 11.00-13.00 (Palazzo Campana, Aula Magna)

Abstract.**A. Villaveces (Bogotá)**"Categoricity, between model theory and set theory".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 behaviour 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.

June 19th, 2015, 11.00-13.00 (Palazzo Campana, Aula 3)

Abstract.**K. Tsaprounis (Salvador de Bahia)**"On ultrahuge cardinals".Starting with the well-known 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 2-hugeness and superhugeness.

June 17th, 2015, 11.00-13.00 (Palazzo Campana, Aula 3)

Abstract.**G. Gherardi (Universität der Bundeswehr München)**"Scommettere aiuta: tra certezze infinite ed errori consapevoli".Mathematical statements of type "For all x in X there exists a y in Y" defines set-theoretically 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 Lebesgue-measurable 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.

June 5th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**L. Nguyen van Thè (Marseille)**"Ramsey-type phenomena from fixed points in compactifications".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 Ramsey-type results.

May 29th, 2015, 11.00-13.00 (Palazzo Campana, Aula 3)

Abstract.**P. Simon (Lyon)**"Order and stability in NIP theories".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 o-minimal theories. The intuition driving its study is that the properties of NIP structures should somehow be a combination of stability and o-minimality. 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 order-like components.

May 27th, 2015, 11.00-13.00 (Palazzo Campana, Aula 2)

Abstract.**D. Asperó (East Anglia)**"Some uses of homogeneous forcing" (Slides).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 $a$?).

May 21st, 2015, 14.00-16.00 (Palazzo Campana, Aula 1)

Abstract.**J. Melleray (Lyon)**"Full groups of minimal homeomorphisms and descriptive set theory".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 descriptive-set-theoretic 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 will try to avoid getting into technical descriptive-set-theoretic discussions.May 15th, 2015, 11.00-13.00 (Palazzo Campana, Aula 3)

Abstract.**H. Mildenberger (Freiburg)**"Subforcings of Blass-Shelah forcing".The second components in Blass--Shelah forcing, the so-called 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 Blass--Shelah 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.

May 8th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**P. Lücke (Bonn)**"Fragments of the Forcing Theorem for Class Forcings" (Slides).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. While 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).

May 4th, 2015, 11.00-13.00 (Palazzo Campana, Aula 1)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 9.April 27th, 2015, 11.00-13.00 (Palazzo Campana, Aula 1)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 8.April 24th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**K. Fournier (Lausanne/Paris)**"Wadge Hierarchy of differences of coanalytic sets".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 co-analytic subsets of the Baire space by extending results obtained by Louveau for the Borel sets.

April 24th, 2015, 11.00-13.00 (Palazzo Campana, Sala S)

Abstract.**Y. Pequignot (Lausanne/Paris)**"A Wadge hierarchy for second countable spaces".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 Hausdorff-Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces.

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 20th, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 7.April 17th, 2015, 11.00-13.00 (Palazzo Campana, Sala S)

Abstract.**V. Gregoriades (Darmstadt)**"Effective descriptive set theory: aspects of the past and directions for the future".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.

April 13th, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 6.April 10th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**L. Newelski (Wrocłav)**"Model theory and topological dynamics".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.

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 model-theoretic properties of groups. Also, the model-theoretic set-up 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 10th, 2015, 11.00-13.00 (Palazzo Campana, Sala S)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 5.March 26th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**A. Chernikov (Paris)**"Action of the automorphism group of a countable omega-categorical structure on its space of types".We discuss topological dynamics of the action of the automorphism group of a countable omega-categorical structure on its space of types. In particular, we consider a definable counterpart of the Kechris-Pestov-Todorcevic correspondence and the effect of various model-theoretic assumptions (stability, NIP, etc).

March 23rd, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 4.March 20th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**V. Mantova (Pisa)**"Surreal numbers, derivations and transseries" (Slides).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' Liouville-chiusa, ovvero surgettiva.

March 20th, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 3.March 9th, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 2.March 6th, 2015, 14.00-16.00 (Palazzo Campana, Aula 3)

Abstract.**A. Mendini (Vienna)**"Dropping polishness" (Slides).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.

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 2nd, 2015, 11.00-13.00 (Palazzo Campana, Aula 1)

**L. Motto Ros (Turin)**"Sulle relazioni di isometria e immersione isometrica tra spazi Polacchi (ultra)metrici", part 1.

TBA

April 2nd, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**D. Marini (Turin)**"The $\mathrm{DDG}$ and very Large Cardinals.", part 2 (Slides).Following the work of Hamkins, Fuchs and Reitz, we study the basic concepts of set-theoretic geology. In particular we recall the definitions of ground model, the downward directed grounds hypothesis $\mathrm{DDG}$, mantle and its ``generalization'', namely the generic mantle. Furthermore, Usuba has shown that the strong $\mathrm{DDG}$ is a theorem of {\tt ZFC}. Consequently, the mantle is a model of {\tt ZFC} and if the universe has some very large cardinal, then the mantle must be a ground. In this case, we are also able to estimate the size of the class of all grounds of the universe $V$.<\br> We present these results sketching the technical proofs and show the relevant consequences.<\br> The talk will be held in Italian.

March 26th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**D. Marini (Turin)**"The $\mathrm{DDG}$ and very Large Cardinals.", part 1 (Slides).Following the work of Hamkins, Fuchs and Reitz, we study the basic concepts of set-theoretic geology. In particular we recall the definitions of ground model, the downward directed grounds hypothesis $\mathrm{DDG}$, mantle and its ``generalization'', namely the generic mantle. Furthermore, Usuba has shown that the strong $\mathrm{DDG}$ is a theorem of {\tt ZFC}. Consequently, the mantle is a model of {\tt ZFC} and if the universe has some very large cardinal, then the mantle must be a ground. In this case, we are also able to estimate the size of the class of all grounds of the universe $V$.<\br> We present these results sketching the technical proofs and show the relevant consequences.<\br> The talk will be held in Italian.

March 1st, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**T. Marinov (Turin)**"A Logic of Strong Contact between Polytopes", part 2.A logic of space is a loose term for a formal system interpreted over a class of structures featuring geometrical entities and relations. The motivations for studying those range from the philosophy of geometry to computer-scientific applications. A main class of structures of interest in that area are the so-called contact algebras - Boolean algebras with a suitable additional binary relation. This seminar will be a presentation of the results in the speaker's Master thesis, which are concerned with a particular newly proposed contact relation. Its intuitive meaning is that two objects are in 'strong contact' if some small enough object with nonzero measure can pass from one to the other without leaving their union. A study of the universal fragments of the theories of a class of the resultant contact algebras will be discussed.

February 22nd, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**T. Marinov (Turin)**"A Logic of Strong Contact between Polytopes", part 1.A logic of space is a loose term for a formal system interpreted over a class of structures featuring geometrical entities and relations. The motivations for studying those range from the philosophy of geometry to computer-scientific applications. A main class of structures of interest in that area are the so-called contact algebras - Boolean algebras with a suitable additional binary relation. This seminar will be a presentation of the results in the speaker's Master thesis, which are concerned with a particular newly proposed contact relation. Its intuitive meaning is that two objects are in 'strong contact' if some small enough object with nonzero measure can pass from one to the other without leaving their union. A study of the universal fragments of the theories of a class of the resultant contact algebras will be discussed.

February 8th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**E. Colla (Turin)**"Some application of model theory to combinatorics", part 3.One of the strongest points of model theory is flexibility. We can use different languages and interpretations to apply essentially the same technique in different contexts. In these seminars we try to give our point of view on the applications of the concept of coheir in combinatorics. Several theorems in combinatorics, e.g. Ramsey's, Hindman's, Carlson's theorems, can be proven in a similar fashion thanks to this concept. We will skip some proof in order to have time to speak about motivation and heuristics, and hopefully to get suggestions on future directions from the audience. The talk is intended for an audience with little background in model theory.

January 25th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**E. Colla (Turin)**"Some application of model theory to combinatorics", part 2.One of the strongest points of model theory is flexibility. We can use different languages and interpretations to apply essentially the same technique in different contexts. In these seminars we try to give our point of view on the applications of the concept of coheir in combinatorics. Several theorems in combinatorics, e.g. Ramsey's, Hindman's, Carlson's theorems, can be proven in a similar fashion thanks to this concept. We will skip some proof in order to have time to speak about motivation and heuristics, and hopefully to get suggestions on future directions from the audience. The talk is intended for an audience with little background in model theory.

January 18th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)

Abstract.**E. Colla (Turin)**"Some application of model theory to combinatorics", part 1.One of the strongest points of model theory is flexibility. We can use different languages and interpretations to apply essentially the same technique in different contexts. In these seminars we try to give our point of view on the applications of the concept of coheir in combinatorics. Several theorems in combinatorics, e.g. Ramsey's, Hindman's, Carlson's theorems, can be proven in a similar fashion thanks to this concept. We will skip some proof in order to have time to speak about motivation and heuristics, and hopefully to get suggestions on future directions from the audience. The talk is intended for an audience with little background in model theory.

November 12th, 2018, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**C. Agostini (Turin)**"Cardinal characteristics of partial orders and $\mathfrak{p}=\mathfrak{t}$", part 3.The continuum Hypothesis CH state the equality between the first uncountable cardinal $\aleph_1$ and the size of the reals $\mathfrak c$. In 1963 Cohen completes the result of Godel proving the independence of CH from ZFC and inventing forcing. These result rise an interest in studying the possible configurations of other cardinals that lies in the interval $[\aleph_1, \mathfrak c]$, but cannot be proven equal nor different from $\aleph_1$ and $\mathfrak c$. These seminars are focused on two of those cardinals that might be regard as the first and most famous ones, $\mathfrak p$ and $\mathfrak t$. We will analyze the steps of the proof provided by Malliaris and Shelah that shows they are equal in every possible model of ZFC. This is done in a slightly different fashion from the original proof. First, we do an analysis of the configuration of analogues of those cardinal characteristics for partial orders in general. Some examples where they can have different configurations follow. Finally, the proof of $\mathfrak p = \mathfrak t$ is provided using the tools previously defined.

November 5th, 2018, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**C. Agostini (Turin)**"Cardinal characteristics of partial orders and $\mathfrak{p}=\mathfrak{t}$", part 2.The continuum Hypothesis CH state the equality between the first uncountable cardinal $\aleph_1$ and the size of the reals $\mathfrak c$. In 1963 Cohen completes the result of Godel proving the independence of CH from ZFC and inventing forcing. These result rise an interest in studying the possible configurations of other cardinals that lies in the interval $[\aleph_1, \mathfrak c]$, but cannot be proven equal nor different from $\aleph_1$ and $\mathfrak c$. These seminars are focused on two of those cardinals that might be regard as the first and most famous ones, $\mathfrak p$ and $\mathfrak t$. We will analyze the steps of the proof provided by Malliaris and Shelah that shows they are equal in every possible model of ZFC. This is done in a slightly different fashion from the original proof. First, we do an analysis of the configuration of analogues of those cardinal characteristics for partial orders in general. Some examples where they can have different configurations follow. Finally, the proof of $\mathfrak p = \mathfrak t$ is provided using the tools previously defined.

October 29th, 2018, 15.30-17.30 (Palazzo Campana, Aula 3)

Abstract.**C. Agostini (Turin)**"Cardinal characteristics of partial orders and $\mathfrak{p}=\mathfrak{t}$", part 1.The continuum Hypothesis CH state the equality between the first uncountable cardinal $\aleph_1$ and the size of the reals $\mathfrak c$. In 1963 Cohen completes the result of Godel proving the independence of CH from ZFC and inventing forcing. These result rise an interest in studying the possible configurations of other cardinals that lies in the interval $[\aleph_1, \mathfrak c]$, but cannot be proven equal nor different from $\aleph_1$ and $\mathfrak c$. These seminars are focused on two of those cardinals that might be regard as the first and most famous ones, $\mathfrak p$ and $\mathfrak t$. We will analyze the steps of the proof provided by Malliaris and Shelah that shows they are equal in every possible model of ZFC. This is done in a slightly different fashion from the original proof. First, we do an analysis of the configuration of analogues of those cardinal characteristics for partial orders in general. Some examples where they can have different configurations follow. Finally, the proof of $\mathfrak p = \mathfrak t$ is provided using the tools previously defined.

March 22nd, 2016, 14.00-16.00 (Palazzo Campana, Aula 4)

**F. Calderoni (Turin)**"Quanto è difficile classificare rappresentazioni unitarie irriducibili?", part 4.March 18th, 2016, 14.30-16.30 (Palazzo Campana)

**F. Calderoni (Turin)**"Quanto è difficile classificare rappresentazioni unitarie irriducibili?", part 3.March 11th, 2016, 14.30-16.30 (Palazzo Campana)

**F. Calderoni (Turin)**"Quanto è difficile classificare rappresentazioni unitarie irriducibili?", part 2.March 4th, 2016, 14.30-16.30 (Palazzo Campana)

**F. Calderoni (Turin)**"Quanto è difficile classificare rappresentazioni unitarie irriducibili?", part 1.February 26th, 2016, 10.30-12.30 (Palazzo Campana, Aula 5)

**G. Audrito (Turin)**"Sistemi di filtri: torri, extender e non solo", part 4.February 19th, 2016, 10.30-12.30 (Palazzo Campana, Aula 5)

**G. Audrito (Turin)**"Sistemi di filtri: torri, extender e non solo", part 3.February 12th, 2016, 10.30-12.30 (Palazzo Campana, Aula 5)

**G. Audrito (Turin)**"Sistemi di filtri: torri, extender e non solo", part 2.January 29th, 2016, 10.30-12.30 (Palazzo Campana, Aula 5)

**G. Audrito (Turin)**"Sistemi di filtri: torri, extender e non solo", part 1.January 15th, 2016, 10.30-12.30 (Palazzo Campana, Aula 3)

**F. Cavallari (Turin)**"Alternating automata and weak alternating automata", part 2.January 8th, 2016, 10.30-12.30 (Palazzo Campana, Aula 3)

**F. Cavallari (Turin)**"Alternating automata and weak alternating automata", part 1.

December 15th, 2015, 08.30-10.30 (Palazzo Campana, Aula 2)

**F. Cavallari (Turin)**"Introduzione ad automi su parole infinite", part 2.December 1st, 2015, 08.30-10.30 (Palazzo Campana, Aula 2)

**F. Cavallari (Turin)**"Introduzione ad automi su parole infinite", part 1.May 29th, 2015, 14.30-16.30 (Palazzo Campana, Aula 3)

Abstract.**H. Nobrega (Amsterdam)**"Game characterizations of functions of finite Baire class".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.

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.May 25th, 2015, 11.00-13.00 (Palazzo Campana, Aula 1)

**F. Calderoni (Turin)**"Preordini analitici completi", part 4.May 20th, 2015, 11.00-13.00 (Palazzo Campana, Aula 2)

**F. Calderoni (Turin)**"Preordini analitici completi", part 3.May 18th, 2015, 11.00-13.00 (Palazzo Campana, Aula 1)

**F. Calderoni (Turin)**"Preordini analitici completi", part 2.May 11th, 2015, 11.00-13.00 (Palazzo Campana, Aula 1)

**F. Calderoni (Turin)**"Preordini analitici completi", part 1.March 30th, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**G. Audrito (Turin)**"Assiomi di resurrezione e risultati di assolutezza generica", part 5.March 27th, 2015, 11.00-13.00 (Palazzo Campana, Aula C)

**G. Audrito (Turin)**"Assiomi di resurrezione e risultati di assolutezza generica", part 4.March 16th, 2015, 11.00-13.00 (Palazzo Campana, Aula 3)

**G. Audrito (Turin)**"Assiomi di resurrezione e risultati di assolutezza generica", part 3.March 13th, 2015, 11.00-13.00 (Palazzo Campana, Sala S)

**G. Audrito (Turin)**"Assiomi di resurrezione e risultati di assolutezza generica", part 2.March 6th, 2015, 11.00-13.00 (Palazzo Campana, Sala S)

**G. Audrito (Turin)**"Assiomi di resurrezione e risultati di assolutezza generica", part 1.January 30th, 2015, 11.00-13.00 (Palazzo Campana)

**G. Carotenuto (Turin)**"Insiemi di densità dei reali", part 3.January 26th, 2015, 11.00-13.00 (Palazzo Campana)

**G. Carotenuto (Turin)**"Insiemi di densità dei reali", part 2.January 21st, 2015, 11.00-13.00 (Palazzo Campana)

**G. Carotenuto (Turin)**"Insiemi di densità dei reali", part 1.January 16th, 2015, 11.00-13.00 (Palazzo Campana)

**S. Steila (Turin)**"Versioni definibili di equivalenze di CH", part 3.January 14th, 2015, 11.00-13.00 (Palazzo Campana)

**S. Steila (Turin)**"Versioni definibili di equivalenze di CH", part 2.January 12th, 2015, 11.00-13.00 (Palazzo Campana)

**S. Steila (Turin)**"Versioni definibili di equivalenze di CH", part 1.

December 12th, 2014, 11.00-13.00 (Palazzo Campana)

**F. Calderoni (Turin)**"Gruppi liberabili", part 3.December 5th, 2014, 11.00-13.00 (Palazzo Campana)

**F. Calderoni (Turin)**"Gruppi liberabili", part 2.December 1st, 2014, 11.00-13.00 (Palazzo Campana)

**F. Calderoni (Turin)**"Gruppi liberabili", part 1.