In 1964, Keisler introduced $\kappa$-good ultrafilters, which can be characterized as those ultrafilters which produce $\kappa$-saturated ultrapowers. The problem of finding an analogous characterization for ultrafilters on Boolean algebras has been considered by Mansfield (1971), Benda (1974), and Balcar and Franek (1982), who proposed and compared different notions of “goodness” for such ultrafilters. In the first part of my talk, I shall outline the different definitions introduced in the literature and show that they are in fact all equivalent, thus providing a complete characterization of those ultrafilters which produce $\kappa$-saturated Boolean ultrapowers. In the second part of the talk, I shall present a joint work with Matteo Viale, which started in 2015 during my Master's thesis and was recently revived during the last few months in Torino. The aim of our project is to study the universality properties of forcing. More precisely, we shall prove that, for many interesting signatures, every model of the universal theory of an initial segment of the universe can be embedded into a model constructed by forcing. To achieve this goal, we build good ultrafilters on forcing notions such as the Lévy collapsing algebra and Woodin's stationary tower.

In this talk we will see how geometric invariants of definable sets in o-minimal structures can be used to understand the structure of groups in several categories.

Doctrines were introduced by Lawvere as an algebraic tool to work with logical theories and their extensions. In fact, this algebraic character makes the theory of doctrines a suitable context where to address the question: "What is the theory obtained by (co)freely adding logical structure?" or the closely related question: "How to express additional logical structure in terms of what is already available?". More precisely, in the first case we ask whether a certain forgetful functor is adjoint and, in the second case, whether the adjunction obtained in this way is (co)monadic. After an introduction to doctrines and their connection to logic and type theory, I shall discuss the above questions in the case of two forgetful functors: the one from theories with conjunctions, equality and quotients to theories with conjunctions and equality, and the one that further forgets equality. Not surprisingly, the answers revolve around the concept of equivalence relation. I shall discuss applications to useful constructions in categorical logic and type theory, as well as to the theory of imaginary elements in the sense of Poizat. If time allows, I shall also describe how to lift this setting to Grothendieck fibrations (of which doctrines are a particular case) using groupoids instead of equivalence relations.

Using tools from the theory of optimal transport, I will discuss a new characterisation of general amenable topological groups. Specifically, let G be an amenable topological group with no non-trivial homomorphisms to R and let d be a left-invariant continuous metric on G. Then one may find finitely supported probability measures on G that are almost invariant, with respect to the Wasserstein distance for the cost function d, under any given finite sets of translations by elements of G. On the other hand, I will also show how this fails in the amenable par excellence group Z. Time permitting, I will also indicate how this can be used to average potentially unbounded Lipschitz functions defined on spaces on which G acts isometrically.

A classical tool in the study of real closed fields are the fields K((G)) of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. A fundamental result of Berarducci ensures the existence of irreducible series in the subring of generalised power series with non-positive exponents. This report aims at describing a factorisation theory in this context: a joint work with Vincenzo Mantova proves that every series admits a factorisation into a bounded number of irreducibles and a unique product, up to multiplication by a unit, of factors whose supports are finite and generate rational linear spaces of dimension one. Analogous results are deduced for the ring of omnific integers within Conway's surreal numbers, using a suitable notion of infinite product. In turn, Gonshor's conjecture is solved: the omnific integer omega 2 + omega + 1 is prime. Other possible generalizations will also be sketched.

Reverse mathematics is a program to classify mathematical theorems by set comprehension axioms in second order arithmetic [1]. In this program, it is presented that most of the theorems from undergraduate mathematics are equivalent to set comprehension axioms characterizing systems called "Big Five". Comparing to the systems of set theory, second order arithmetic is a rather weak system, which enables the classification of weak determinacy schemata for the classes in the very low level of the Wadge hierarchy. In this talk, we will see that determinacy of infinite games up to the defference hierarchy over \Sigma^0_3 makes a fine hierarchy in second order arithmetic. References [1] S. G. Simpson, Subsystems of second order arithmetic (2nd edition), Cambridge University Press, 2010 [2] T. Nemoto, Determinacy of Wadge classes and subsystems of second order arithmetic, Mathematical Logic Quarterly, Volume 55, Issue 2, February 2009, pp. 154 - 176. [3] A. Montalbán and R. A. Shore, The limits of determinacy in second order arithmetic: consistency and complexity strength, Israel J. Math., 204 (2014), 477--508.

In this talk I discuss generalizations of the classic finite Ramsey theorem that substitute "set of cardinality n" with the notion of alpha-large set, where alpha is a countable ordinal. The prototype of these results is the statement that Paris and Harrington showed unprovable in PA in 1977. Since then several extensions were proved, typically for ordinals up to epsilon_0. Our results extend this approach by dealing with ordinals (at least) up to Gamma_0 and using simultaneously alpha-large sets (almost) everywhere in the statements. Quite surprisingly, in many cases we obtain tight bounds on the generalized Ramsey numbers, in contrast with the classical finite case where tight bounds are known only for very few cases involving very small numbers. This is joint work with Antonio Montalbán.

Blanchard introduced the concepts of Uniform Positive Entropy (UPE) and Complete Positive Entropy (CPE) as topological analogues of K-automorphism. He showed that UPE implies CPE, and that the converse is false. A flurry of recent activity studies the relationship between these two notions. For example, one can assign a countable ordinal which measures how complicated a CPE system is. Recently, Barbieri and García-Ramos constructed Cantor CPE systems at every level of CPE. Westrick showed that natural rank associated to CPE systems is actually a \Pi^1_1-rank. More importantly, she showed that the collection of CPE Z2-SFT's is a \Pi^1_1-complete set. In this talk, we discuss some results, where UPE and CPE coincide and others where we show that the complexity of certain classes of CPE systems is \Pi^1_1-complete. This is joint work with García-Ramos.

A Borel action of a Polish locally compact group on a standard Borel space admits a countable Borel section, i.e., a Borel set that meets every orbit in a countable nonempty set. It is a long standing open problem whether this property characterizes locally compact groups. I will discuss the history of this problem and some recent progress in joint work with M. Malicki, A. Panagiotopoulos and J. Zielinski.

In a paper published in 2008, Macpherson and Steinhorn introduced and studied structures in which each every definable set carries a well behaved dimension and measure: we refer to these as MS-measurable structures. Examples include totally categorical structures, pseudofinite fields and the random graph. MS-measurable structures are supersimple of finite SU-rank and we discuss some amalgamation properties which hold in MS-measurable structures, but not in all supersimple finite rank structures. We are interested in the question of whether every omega-categorical, MS-measurable structure is one-based. A construction of Hrushovski can be used to produce omega-categorical structures which are supersimple of finite SU-rank and not one-based: indeed, this construction is essentially the only known way to produce such structures. It is still an open question whether any of these Hrushovski constructions can be MS-measurable. However, I will discuss some work of myself and of my PhD student Paolo Marimon which uses the amalgamation results and other methods to show that at least some of the Hrushovski constructions are not MS-measurable.

In his seminal work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals known as the Magidor iteration. The purpose of this talk is to present a Mathias-type criterion which characterizes when a sequence of omega-sequences is generic for the Magidor iteration. The result extends a theorem of Fuchs who introduced a Mathias criterion for discrete products of Prikry forcings. We will present the new criterion, discuss several applications, and outline the main ideas of the proof

I will give a brief introduction to the program of reverse mathematics, which seeks to answer the ancient question of which mathematical axioms are necessary to prove theorems of ordinary mathematics. I will then discuss a special theorem of combinatorics, Ramsey’s theorem, which has played an important role in this endeavor, and led to a tantalizing problem known as the SRT22 vs. COH problem. I will focus on this problem, talk about its history, and then briefly discuss its recent solution by Monin and Patey. I hope for the talk to be accessible to a general mathematical audience.

There are several statements in elementary geometry that depend on the size of the continuum, and most of them are modelled on the proof of a theorem of Sierpiński's. In the first part of the talk I will survey a few of these geometric statements and show how these are related to each other. In the second part I will show how imposing a definability condition on the pieces of Sierpiński's theorem yields a better bound on the size of the continuum.

The infinite binary tree (i.e. the free structure of two successors, aka S2S) seems to be a very simple and natural object. Nevertheless, due to its branching structure, it has rich abilities of modelling complex processes including e.g. nondeterminism, perfect information games, combinatorics of P(N), etc The fundamental result of Rabin from late 60's (sometimes called ""the mother of all decidability results"") proves that the Monadic Second-Order (MSO) theory of S2S is decidable. Since then, the structure of properties expressible in MSO over S2S has been intensively studied. Many of these studies were related to and/or motivated by descriptive set theory. During the talk I would like to make a broad overview of these relations, including issues of Wadge degrees, measurability (with relations to Kolmogorov), and uniformisability (Gurevich-Shelah). Although a lot of questions have been already answered, there still remain important and natural open problems in all three mentioned directions of research.

A “pathological set” can be a non measurable set, a set which does not have the property of Baire (namely it is not a Borel set modulo a first category set).
A subset (=the infinite subsets of natural numbers) can be considered to be ”pathological” if it is a counterexample to the infinitary Ramsey theorem. Namely there does not exist an infinite set of natural numbers such that all its infinite subsets are in our sets or all its infinite subsets are not in the set.
A subset of the Baire space can be considered to be “pathological” if the infinite game is not determined. The game is an infinite game where two players alternate picking natural numbers, forming an infinite sequence, namely a member of . The first player wins the round if the resulting sequence is in . The game is determined if one of the players has a winning strategy.
A prevailing paradigm in Descriptive Set Theory is that sets that have a “simple description” should not be pathological. Evidence for this maxim is the fact that Borel sets are not pathological in any of the senses above.In this talk we shall present a notion of “super regularity” for subsets of a Polish space, the family of universally Baire sets. This family of sets generalizes the family of Borel sets and forms a -algebra. We shall survey some regularity properties of universally Baire sets , such as their measurability with respect to any regular Borel measure, the fact that they have an infinitary Ramsey property etc. Some of these theorems will require assuming some strong axioms of infinity. Most of the talk should be accessible to a general Mathematical audience, but in the second part we shall survey some newer results.

There are several familiar theories of reverse mathematics that can be characterized by well-ordering principles of the form (*) "if $X$ is well ordered then $f(X)$ is well ordered", where $f$ is a standard proof theoretic function from ordinals to ordinals (such $f$'s are always dilators). Some of these equivalences have been obtained by recursion-theoretic and combinatorial methods. They (and many more) can also be shown by proof-theoretic methods, employing search trees and cut elimination theorems in infinitary logic with ordinal bounds. One could perhaps generalize and say that every cut elimination theorem in ordinal-theoretic proof theory encapsulates a theorem of this type.

The Schröder-Bernstein theorem asserts that if there are injections of two sets into each other, then there is a bijection between the sets. In his note "Un théorème sur les transformations biunivoques," Banach proved an extension of the Schröder-Bernstein theorem in which the values of the bijection between the sets depend directly on the injections. This talk will present some old theorems of reverse mathematics about restrictions of Banach's theorem. Also, we will look at preliminary results of work with Carl Mummert on restrictions of Banach's theorem in higher order reverse mathematics. The talk will not assume familiarity with reverse mathematics.

The method of forcing was introduced by Cohen in his proof of the independence of the Continuum Hypothesis and has since been used to demonstrate that a diverse array of set theoretic problems are formally unsolvable from the standard ZFC axioms. The technique allows one to expand a model of ZFC by adjoining to it a generic set. The resulting forcing extension is again a model of ZFC that may have a very different first order theory from the original structure; for example, according to one's tastes, one can build forcing extensions in which the Continuum Hypothesis is either true or false, demonstrating that the ZFC axioms can neither prove nor refute the Continuum Hypothesis. But does the forcing technique really show that the Continuum Hypothesis has no truth value? This seems to hinge on whether one believes that the true universe of sets (which the ZFC axioms attempt to axiomatize) could itself be a forcing extension of a smaller model of ZFC. This talk concerns a theorem of Usuba that bears on this question. I'll discuss recent work proving the optimality of the large cardinal hypothesis of Usuba's theorem and some applications of the associated techniques to questions outside the theory of forcing.

Given an abelian group \(G\), a corner is a subset of pairs of the form \(\{ (x,y), (x+g, y), (x, y+g)\}\) with \(g\) non trivial. Ajtai and Szemerédi proved that, asymptotically for finite abelian groups, every dense subset \(S\) of \(G\times G\) contains a corner. Shkredov gave a quantitative lower bound on the density of the subset \(S\). In this talk, we will explain how model-theoretic conditions on the subset \(S\), such as local stability, will imply the existence of corners and of cubes for (pseudo-)finite abelian groups. This is joint work with D. Palacin (Madrid) and J. Wolf (Cambridge).

Jens Mammen (Professor Emeritus of psychology at Aarhus and Aalborg University) has developed a theory in psychology, which aims to provide a model for the interface between a human being (and mind), and the real world.
This theory is formalized in a very mathematical way: Indeed, it is described through a mathematical axiom system. Realizations ("models") of this axiom system consist of a non-empty set $U$ (the universe of objects), as well as a perfect Hausdorff topology $\mathcal{S}$ on $U$, and a family $\mathcal{C}$ of subsets of $U$ which must satisfy certain axioms in relation to $\mathcal{S}$. The topology $\mathcal{S}$ is used to model broad categories that we sense in the world (e.g., all the stones on a beach) and the $\mathcal{C}$ is used to model the process of selecting an object in a category that we sense (e.g., a specific stone on the beach that we pick up). The most desirable kind of model of Mammen's theory is one in which every subset of $U$ is the union of an open set in $\mathcal{S}$ and a set in $\mathcal{C}$. Such a model is called "complete".
Coming from mathematics, models of Mammen's theory were first studied in detail by J. Hoffmann-Joergensen in the 1990s. Hoffmann-Joergensen used the Axiom of Choice (AC) to show that a complete model of Mammen's axiom system, in which the universe $U$ is infinite, does exist. Hoffmann-Joergensen conjectured at the time that the existence of a complete model of Mammen's axioms would imply the Axiom of Choice.
In this talk, I will discuss various set-theoretic aspects related to complete Mammen models; firstly, the question of "how much" AC is needed to obtain a complete Mammen model; secondly, I will introduce some cardinal invariants related to complete Mammen models and establish elementary ZFC bounds for them, as well as some consistency results.
This is joint work with Jens Mammen.

Determinacy assumptions, large cardinal axioms, and their consequences are widely used and have many fruitful implications in set theory and even in other areas of mathematics. Many applications, in particular, proofs of consistency strength lower bounds, exploit the interplay of determinacy axioms, large cardinals, and inner models. I will survey some recent results in this flourishing area. This, in particular, includes results on connecting the determinacy of longer games to canonical inner models with many Woodin cardinals, a new lower bound for a combinatorial statement about infinite trees, as well as an application of determinacy answering a question in general topology.

In structural Ramsey theory, one considers a "small" structure A, a "medium" structure B, a "large" structure C and a number r, then considers the following combinatorial question: given a coloring of the copies of A inside C in r colors, can we find a copy of B inside C all of whose copies of A receive just one color? For example, when C is the rational linear order and A and B are finite linear orders, then this follows from the finite version of the classical Ramsey theorem. More generally, when C is the Fraisse limit of a free amalgamation class in a finite relational language, then for any finite A and B in the given class, this can be done by a celebrated theorem of Nesetril and Rodl. Things get much more interesting when both B and C are infinite. For example, when B and C are the rational linear order and A is the two-element linear order, a pathological coloring due to Sierpinski shows that this cannot be done. However, if we weaken our demands and only ask for a copy of B inside C whose copies of A receive "few" colors, rather than just one color, we can succeed. For the two-element linear order, we can get down to two colors. For the three-element order, 16 colors. This number of colors is called the big Ramsey degree of a finite structure in a Fraisse class. Recently, building on groundbreaking work of Dobrinen, I proved a generalization of the Nešetril-Rödl theorem to binary free amalgamation classes defined by a finite forbidden set of irreducible structures (for instance, the class of triangle-free graphs), showing that every structure in every such class has a finite big Ramsey degree. My work only bounded the big Ramsey degrees, and left open what the exact values were. In recent joint work with Balko, Chodounský, Dobrinen, Hubicka, Konecný, and Vena, we characterize the exact big Ramsey degree of every structure in every binary free amalgamation class defined by a finite forbidden set.

We will recall the notion of compact and hereditary families of finite subsets of some cardinal $\kappa$ and their corresponding combinatorial Banach spaces. We present a combinatorial version of Banach-Stone theorem, which leads naturally to a notion of isomorphism between families. Our main result shows that different families on $\omega$ are not isomorphic, if we assume them to be spreading. We also discuss the difference between the countable and the uncountable setting. This is a joint work with Claribet Piña.

The idea of defining a generic version of a large cardinal by asking that some form of the elementary embeddings characterizing the large cardinal exist in a forcing extension has a long history. A large cardinal (typically measurable or stronger) can give rise to several natural generic versions with vastly different properties. For a \emph{generic large cardinal}, a forcing extension should have an elementary embedding $j:V\to M$ of the form characterizing the large cardinal where the target model $M$ is an inner model of the forcing extension, not necessarily contained in $V$. The closure properties on $M$ must correspondingly be taken with respect to the forcing extension. Very small cardinals such as $\omega_1$ can be generic large cardinals under this definition. Quite recently set theorists started studying a different version of generic-type large cardinals, called \emph{virtual large cardinals}. Large cardinals characterized by the existence of an elementary embedding $j:V\to M$ typically have equivalent characterizations in terms of the existence of set-sized embeddings of the form $j:V_\lambda\to M$. For a virtual large cardinal, a forcing should have an elementary embedding $j:V_\lambda\to M$ of the form characterizing the large cardinal with $M\in V$ and all closure properties on $M$ considered from $V$'s standpoint. Virtual large cardinals are actually large cardinals, they are completely ineffable and more, but usually bounded above by an $\omega$-Erdös cardinal. Despite sitting much lower in the large cardinal hierarchy, they mimic the reflecting properties of their original counterparts. Several of these notions arose naturally out of equiconsistency results. In this talk, I will give an overview of the virtual large cardinal hierarchy including some surprising recent directions.

"The automorphism group Aut(A) of a countable countably categorical structure A, viewed as a topological group equipped with the topology of pointwise convergence, carries sufficient information about the structure A to reconstruct it up to bi-interpretability. It turns out that in many cases, including the order of the rational numbers or the random graph, the algebraic group structure of Aut(A) alone is sufficient for this kind of reconstruction, since its topology is already uniquely determined by it. Which structures A have this property has been subject to investigations for many years. Sometimes, we wish to associate to the structure A other objects than Aut(A) which retain more information about A; for example, its endomorphism monoid End(A) or its polymorphism clone Pol(A) are such objects. As in the case for automorphism groups, these objects are naturally equipped with the topology of pointwise convergence on top of their algebraic structure. We consider the question of when the former is already uniquely determined by the latter. In particular, we show that the endomorphism monoid of the random graph has a unique Polish topology, namely that of pointwise convergence. In the first part of the talk, which I hope to make accessible to anyone, I present a history of the known and unknown results as well as our new ones, and outline the differences between groups and monoids in this context. In the second part, which I also hope to make accessible to anyone, I try to outline the proof methods for our new results. This is joint work with L. Elliott, J. Jonušas, J. D. Mitchell, and Y. Péresse."

We will investigate an interplay between short exact sequences of topological groups and their universal minimal flows in case one of the factors is compact. We will discuss possible and impossible extensions of the results in a few directions. An indispensable ingredient in our technique is a description of the universal pointed flow of a given group in terms of filters on the group, which we will describe.

Of the principles just slightly weaker than ATR, the most well-known are the theories of hyperarithmetic analysis (THA). By definition, such principles hold in HYP. Motivated by the question of whether the Borel Dual Ramsey Theorem is a THA, we consider several theorems involving Borel sets and ask whether they hold in HYP. To make sense of Borel sets without ATR, we formalize the theorems using completely determined Borel sets. We characterize the completely determined Borel subsets of HYP as precisely the sets of reals which are $\Delta^1_1$ in $L_{\omega_1^\mathrm{CK}}$. Using this, we show that in HYP, Borel sets behave quite differently than in reality. In HYP, the Borel dual Ramsey theorem fails, every $n$-regular Borel acyclic graph has a Borel $2$-coloring, and the prisoners have a Borel winning strategy in the infinite prisoner hat game. Thus the negations of these statements are not THA. Joint work with Henry Towsner and Rose Weisshaar.

After Sela and Kharlampovich-Myasnikov proved that nonabelian free groups share the same common theory, a model theoretic interest for the theory of the free group arose. Moreover, maybe surprisingly, Sela proved that this common theory is stable. Stability is the first dividing line in Shelah's classification theory and it is equivalent to the existence of a nicely behaved independence relation - forking independence. This relation, in the theory of the free group, has been proved (Ould Houcine-Tent and Sklinos) to be as complicated as possible (n-ample for all n). This behavior of forking independence is usually witnessed by the existence of an infinite field. We prove that no infinite field is interpretable in the theory of the free group, giving the first example of a stable group which is ample but does not interpret an infinite field.

We explore the uniform computational strength of the problem DS of computing an infinite descending sequence through an ill-founded linear order. This is done by characterizing its degree from the point of view of Weihrauch reducibility, and comparing it with the one of other classical problems, like the problem of finding a path through an ill-founded tree (known as choice on the Baire space). We show that, despite being ""hard"" to compute, the lower cone of DS misses many arithmetical problems (in particular, DS uniformly computes only the limit computable functions). We also generalize our results in the context of arithmetically or analytically presented quasi orders. In particular, we use a technique based on inseparable $\Pi^1_1$ sets to separate $\Sigma^1_1$-DS from the choice on Baire space.

A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit, the countable homogeneous universal Steiner triple system M. In joint work with Enrique Casanovas, we have proved that the theory T of M has quantifier elimination, is not small, has TP2, NSOP1, eliminates hyperimaginaries and weakly eliminates imaginaries. In this talk I will review the construction of M, give an axiomatisation of T and prove some of its properties.

We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.

"In this talk we will give an overview of the theory of \Sigma-Prikry forcings and their iterations, recently introduced in a series of papers. We will begin motivating the class of \Sigma-Prikry forcings and showing that this class is broad enough to encompass many Prikry-type posets that center on countable cofinalities. Afterwards, we will present a viable iteration scheme for this family and discuss an application of the framework to the investigation of stationary reflection at the level of successors of singular cardinals. This is joint work with A. Rinot and D. Sinapova."

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such that whenever $f$ and $g$ are in the set, $f+g$, $fg$ and $f^g$ are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^x}$. They did so by studying the possible limits at infinity of the quotient $f(x)/g(x)$ of two functions in the fragment: if $g$ is kept fixed and $f$ varies, the possible limits form a discrete set of real numbers of order type $\omega$. Using the surreal numbers, we extend the latter result to the whole class of Skolem functions and we discuss some additional progress towards the conjecture of Skolem. This is joint work with Marcello Mamino (http://arxiv.org/abs/1911.07576, to appear in the JSL).

In this seminar we want to take stock of some of the most important applications of the peculiarities of Prikry-like forcings on generalized descriptive set theory. In our case, with generalized descriptive set theory we mean the study of definable subsets of $\lambda^\omega$, with $\lambda$ uncountable cardinal of countable cofinality. It turns out that in this case there is a lot of symmetry with the classical case of Polish spaces, and we are going to provide three examples where the particular combinatorial structure of Prikry-like forcings comes in to save the day: an adequate definition of $\lambda$-Baire property for the generalized case, a generic absoluteness result under the very large cardinal I0, and the construction of a Solovay-like model for $\lambda^\omega$, i.e., the construction of a model where each subset of $\lambda^\omega$ either has cardinality less or equal then $\lambda$, or we can embed in it the whole $\lambda^\omega$.

We prove that the Borel space of torsion-free Abelian groups with domain $\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.

We say that a subset A of the sphere r-divides it if r-many rotations of A perfectly tile the sphere's surface. Such divisions were first exhibited by Robinson (47) and developed by Mycielski (55). We discuss a colorful approach to finding these divisions which are Lebesgue measurable or possess the property of Baire. This includes joint work with J. Grebik, A. Marks, O. Pikhurko, and S. Unger.

I will present a dichotomy for equivalence relations on Polish spaces that can be expressed as countable unions of smooth Borel equivalence relations. It can be seen as an extension of Kechris-Louveau's dichotomy for hypersmooth Borel equivalence relations. A generalization of our dichotomy, for equivalence relations that can be expressed as countable unions of Borel equivalence relations belonging to certain fixed classes, will also be presented. This is a joint work with Benjamin Miller.

A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit, the countable homogeneous universal Steiner triple system M. In joint work with Enrique Casanovas, we have proved that the theory T of M has quantifier elimination, is not small, has TP2, NSOP1, eliminates hyperimaginaries and weakly eliminates imaginaries. In this talk I will review the construction of M, give an axiomatisation of T and prove some of its properties.

Recall Ramsey's theorem for pairs and two colors, which, in terms of graphs, may be phrased as follows: For every countably infinite graph $G$, there is an infinite set of vertices $H$ such that either every pair of distinct vertices from $H$ is adjacent or no pair of distinct vertices from $H$ is adjacent. The conclusion of Ramsey's theorem gives complete information about how the vertices in $H$ relate to each other, but it gives no information about how the vertices outside $H$ relate to the vertices inside $H$. Rival and Sands (1980) proved the following theorem, which weakens the conclusion of Ramsey's theorem with respect to pairs of vertices in $H$, but does add information about how the vertices outside $H$ relate to the vertices inside $H$: For every countably infinite graph $G$, there is an infinite set of vertices $H$ such that each vertex of $G$ is either adjacent to no vertices of $H$, to exactly one vertex of $H$, or to infinitely many vertices of $H$. We give an exact characterization of the computational strength of the Rival-Sands theorem by showing that it is strongly Weihrauch equivalent to the double-jump of weak König's lemma (which is the problem of producing infinite paths through infinite trees that are given by $\Delta^0_3$ approximations). In terms of Ramsey's theorem, this means that solving one instance of the Rival-Sands theorem is equivalent to simultaneously solving countably many instances of Ramsey's theorem for pairs and two colors in parallel. This work is joint with Marta Fiori Carones and Giovanni Soldà.

"A number of well-studied properties of Polish groups concern the interactions between the topological and algebraic structure of those groups. Examples of such properties are the small index property, the automatic continuity, and the Bergman property. An important approach for proving them is showing that the group has ample generics. Therefore we are often interested whether a given Polish group has a comeager conjugacy class, i.e a generic element, a generic pair, or more generally, a generic n-tuple. After a survey on this topic, I will discuss a recent result joint with Aristotelis Panagiotopoulos, where we show that the automorphism group of the random poset does not admit a generic pair. This answers a question of Truss and Kuske-Truss."

We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a -property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. Part (but not all) of our results are a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.

Families of infinite sets of natural numbers are said to be independent if for very two disjoint non-empty subfamilies the intersection of the members of the first subfamily with the complements of the members of the second family is infinite. Maximal independent families are independent families which are maximal under inclusion. In this talk, we will consider the set of cardinalities of maximal independent families, referred to as the spectrum of independence, and show that this set can be quite arbitrary. This is a joint work with Saharon Shelah.

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and ""consistent"" needs to mean ""consistent in a strong sense."" It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler's ""consistency properties.""

In this talk, I show how Goodstein's classical theorem can be turned into a statement that entails the existence of complex infinite sets, or in other words: into an object of reverse mathematics. This more abstract approach allows for very uniform results of high explanatory power. Specifically, I present versions of Goodstein's theorem that are equivalent to arithmetical comprehension and arithmetical transfinite recursion. To approach the latter, we will study a functorial extension of the Ackermann function to all ordinals. The talk is based on a joint paper with J. Aguilera, M. Rathjen and A. Weiermann.

In this talk we shall present some results on left-orderable groups and their interplay with descriptive set theory. We shall discuss how Borel classification can be used to analyze the space of left-orderings of a given countable group modulo the conjugacy action. In particular, we shall see that if G is not locally indicable then the conjugacy relation on LO(G) is not smooth. Also, if G is a nonabelian free group, then the conjugacy relation on LO(G) is a universal countable Borel equivalence relation. Our results address a question of Deroin-Navas-Rivas and show that in many cases LO(G) modulo the conjugacy action is nonstandard. This is joint work with A. Clay.

Can we have both a saturated ideal and the tree property on $\aleph_2$? Towards the negative direction, we show that for a regular cardinal $\kappa$, if $2^{<\kappa}\leq\kappa^+$ and there is a weakly presaturated ideal on $\kappa^+$ concentrating on cofinality $\kappa$, then $\square^*_\kappa$ holds. This partially answers a question of Foreman and Magidor about the approachability ideal on $\aleph_2$. A surprising corollary is that if there is a presaturated ideal $J$ on $\aleph_2$ such that $P(\aleph_2)/J$ is a semiproper forcing, then CH holds. This is joint work with Sean Cox.

We study the isomorphism relation for countable ordered Archimedean groups. We locate its complexity with respect to the hierarchy defined by Hjorth, Kechris, and Louveau, showing in particular that its potential complexity is $\mathrm{D}(\mathbf{\Pi}^0_3)$, and it cannot be classified using countable sets of reals as invariants. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups and ordered divisible Abelian groups. This is joint work with F. Calderoni, D. Marker, and L. Motto Ros.

The study of word problems dates back to the work of Dehn in 1911. Given a recursively presented algebra $A$, the word problem of $A$ is to decide if two words in the generators of $A$ refer to the same element. Nowadays, much is known about the complexity of word problems for algebraic structures: e.g., the Novikov-Boone theorem – one of the most spectacular applications of computability to general mathematics – states that the word problem for finitely presented groups is unsolvable. Yet, the computability theoretic tools commonly employed to measure the complexity of word problems (Turing or m-reducibility) are defined for sets, while it is generally acknowledged that many computational facets of word problems emerge only if one interprets them as equivalence relations. In this work, we revisit the world of word problems through the lens of the theory of equivalence relations, which has grown immensely in recent decades. To do so, we employ computable reducibility, a natural effectivization of Borel reducibility. This is joint work with Valentino Delle Rose and Andrea Sorbi.

We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a property formalized in an appropriate language for second or third order number theory is forcible from some $T$ extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of $T$. Part (but not all) of our results are a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.

Many notions of large cardinals have associated ideals, and also operators on ideals. Classical examples of this are the subtle, the ineffable, the pre-Ramsey and the Ramsey operator. We will recall their definitions, and show that they can be seen to fit within a framework for large cardinal operators below measurability. We will use this framework to introduce a new operator, that is closely connected to the notion of a $T_\omega^\kappa$-Ramsey cardinal that was recently introduced by Philipp Luecke and myself, and we will provide a sample result about our framework that generalizes classical results of James Baumgartner.

This talk is a survey on large cardinals and the relations between elementary embeddings and ultrafilters.

A measurable cardinal is a cardinal $\kappa$ such that there exits a non-principal, $\kappa$-complete ultrafilter on $\kappa$. This has been the first large cardinal defined through ultrafilters. Measurable cardinals however have also an alternative definition: a cardinal is measurable if and only if it is the critical point of a (definable) elementary embedding of the universe into an inner model of set theory. Historically, this second definition has proven more suitable to define large cardinals. One may require that the elementary embedding satisfy some additional properties to obtain even larger cardinals, and measurable cardinals soon became the smallest of a series of large cardinals defined through elementary embeddings. Later on, a definition with ultrafilters has been discovered for many of these cardinals. Similarly to what happens for measurable cardinals, every elementary embedding generates an ultrafilter, and every ultrafilter generates an elementary embedding through the ultrapower of the universe. In this duality, properties of ultrafilters translate into properties of the embeddings, and vice versa. This lead in particular to the definition of two new crucial properties of ultrafilters: fineness and normality.

In the first part of this talk, we will focus on the analysis and comprehension of these two properties of ultrafilters. In the second part, we will focus on the understanding of the "duality" between elementary embeddings and ultrafilters, with a particular attention on the ultrafilters/elementary embedding generated by Supercompact and Huge cardinals.

This talk is a survey on large cardinals and the relations between elementary embeddings and ultrafilters.

A measurable cardinal is a cardinal $\kappa$ such that there exits a non-principal, $\kappa$-complete ultrafilter on $\kappa$. This has been the first large cardinal defined through ultrafilters. Measurable cardinals however have also an alternative definition: a cardinal is measurable if and only if it is the critical point of a (definable) elementary embedding of the universe into an inner model of set theory. Historically, this second definition has proven more suitable to define large cardinals. One may require that the elementary embedding satisfy some additional properties to obtain even larger cardinals, and measurable cardinals soon became the smallest of a series of large cardinals defined through elementary embeddings. Later on, a definition with ultrafilters has been discovered for many of these cardinals. Similarly to what happens for measurable cardinals, every elementary embedding generates an ultrafilter, and every ultrafilter generates an elementary embedding through the ultrapower of the universe. In this duality, properties of ultrafilters translate into properties of the embeddings, and vice versa. This lead in particular to the definition of two new crucial properties of ultrafilters: fineness and normality.

In the first part of this talk, we will focus on the analysis and comprehension of these two properties of ultrafilters. In the second part, we will focus on the understanding of the "duality" between elementary embeddings and ultrafilters, with a particular attention on the ultrafilters/elementary embedding generated by Supercompact and Huge cardinals.

The concept of metastable convergence was introduced by Terry Tao as a tool for his 2008 ergodic theorem. It turns out that this concept is intimately related to ideas that had been used by logicians for decades. Moreover, it arises naturally in many other areas of mathematics, and it connects different subareas of logic in unexpected ways. Note: This is a reprise of seminars given in Firenze and in late February.

Semialgebraic groups over a real closed field can be seen as a generalization of the semialgebraic groups over the real field, and also as a particular case of the groups definable in an o-minimal structure.

In this talk, I will offer a description of the definably connected semialgebraic groups over a real closed field $R$ through the study of their o-minimal universal covering groups and of their relationship with the $R$-points of some connected $R$-algebraic group.

I will show that the o-minimal universal covering group of a definably compact definably connected group definable in a sufficiently saturated real closed field $R$ is an open subgroup of the o-minimal universal covering group of the $R$-points $H(R)$ of some Zariski-connected $R$-algebraic group $H$.

This research is part of my PhD thesis at the University of Haifa, Israel and Universidad de los Andes, Colombia.

Singular cardinals yield surprising results in set theory. After Cohen proved that $\textsf{CH}$ is independent of $\textsf{ZFC}$, Easton proved that on regular cardinals, the continuum function $\kappa \mapsto 2^\kappa$ is constrained only by the facts that $\lambda \le \kappa \Rightarrow 2^\lambda \le 2^\kappa$ and that $\operatorname{cf}(2^\kappa)>\kappa$. In other words, the $\textsf{ZFC}$ constraints on $\kappa \mapsto 2^\kappa$ are fully characterized relative to the class of regular cardinals. In an unexpected turn, Silver proved that $\textsf{GCH}$ cannot fail for the first time at a singular cardinal of uncountable cofinality. Hence, the failure of $\textsf{CH}_\kappa$ is compact for such cardinals.

We are not limited to studying cardinal arithmetic when considering these compactness phenomena. One can also investigate the compactness of the combinatorial properties that characterize Gödel's Constructible Universe $L$. In this way, we can get a sense of which deviations from $L$-like behavior are consistent with $\textsf{ZFC}$. For example, in $L$ every successor cardinal $\kappa^+$ carries a non-reflecting stationary set, and every inaccessible cardinal $\mu$ carries a non-reflecting stationary set if and only if $\mu$ is not weakly compact. But we can ask: which other patterns are possible?

To use more formal language: if $S$ is a stationary subset of a cardinal $\kappa$, the reflection principle $\textsf{SR}(S)$ asserts that every stationary subset of $S$ reflects. As joint work with Sy-David Friedman, we obtained an Easton-style result demonstrating that if we fix some regular $\theta$, there are only a few trivial $\textsf{ZFC}$ constraints on the behavior of $\textsf{SR}(\kappa \cap \operatorname{cof}(\theta))$ as $\kappa$ varies. For this talk, we will present the most challenging cases for this result, which involve the successors of singular cardinals. If time permits, we will discuss tentative steps towards addressing the analogous question for Jensen's $\square_\kappa$, which is joint work with both Sy-David Friedman and Dima Sinapova.

We show that set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship.

We present the following two results:

(1) The theory ZFC$_\omega$ (definable extension of ZFC in the signature $\sigma_\omega$ admitting predicates for $\Delta_0$-formulae and projective sets of reals) has a model companion (which is the theory ZFC-“power-set”+”all sets are countable”).

(2) For any $T$ extending ZFC$_\omega + $“there are class many Woodin cardinals “ and for any $\Pi_2$-sentence $\psi$ in the signature $\sigma_\omega$, the following are equivalent:
(A) $S_\forall+\psi$ is consistent for all $S$ extending $T$ (where $S_\forall$ is the family of $\Pi_1$-sentences provable from $S$);
(B) $T$ proves $\psi^{H_{\omega_1}}$;
(C) $T$ proves that $\psi^{H_{\omega_1}}$ is forcible.

We have results of the same vein also for expansion of set theory which include predicates for the non-stationary ideal on $\omega_1$. Details will be given in the seminar.

We show that set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship.

We present the following two results:

(1) The theory ZFC$_\omega$ (definable extension of ZFC in the signature $\sigma_\omega$ admitting predicates for $\Delta_0$-formulae and projective sets of reals) has a model companion (which is the theory ZFC-“power-set”+”all sets are countable”).

(2) For any $T$ extending ZFC$_\omega + $“there are class many Woodin cardinals “ and for any $\Pi_2$-sentence $\psi$ in the signature $\sigma_\omega$, the following are equivalent:
(A) $S_\forall+\psi$ is consistent for all $S$ extending $T$ (where $S_\forall$ is the family of $\Pi_1$-sentences provable from $S$);
(B) $T$ proves $\psi^{H_{\omega_1}}$;
(C) $T$ proves that $\psi^{H_{\omega_1}}$ is forcible.

We have results of the same vein also for expansion of set theory which include predicates for the non-stationary ideal on $\omega_1$. Details will be given in the seminar.

The automorphism groups of countable homogeneous structures are natural examples of separable and completely metrizable topological groups. The richness of their topological properties have recently brought to light a crucial interplay between Fraïssé amalgamation theory and other areas of mathematics such as Ramsey theory and topological dynamics. Moreover they have been studied extensively as permutation groups. In this talk we focus on the latter aspect. We will discuss how certain model theoretic properties are used to analyze the normal subgroup structure of a large class of those groups. In particular, will see that if M is the order expansion of the Fraïssé limit of a free, transitive and nontrivial amalgamation class, then Aut(M) is simple. This is joint work with Kwiatkowska and Tent.

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 here

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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 $(M,+)$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

TBA

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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...

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

I am going to explain the notion of metric ultraproduct of structures (mainly groups) and give applications.

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.

We will present some operational set theories (introduced by Feferman) and compare their strength with classical set theories.

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.

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 $C^+(X,Y)$.

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 $C^+(X,Y)$.

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)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $a$?).

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.

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.

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).

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.

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.

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.

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.

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).

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.

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.

This is joint work with Matteo Viale. It is well known that the completeness theorem for $L_{\omega_1\omega}$ fails with respect to Tarski semantics. Mansfield showed that it holds for $L_{\kappa\kappa}$ if one replaces Tarski semantics with boolean valued semantics. I'll talk about using forcing to improve his result in order to obtain a stronger form of boolean completeness (but only for $L_{\kappa\omega}$). Leveraging on this completeness result, one can establish the Craig interpolation property and a strong version of the omitting types theorem for $L_{\kappa\omega}$ with respect to boolean valued semantics. I'll also present a weak version of these results for the general case $L_{\kappa\lambda}$ (if one leverages instead on Mansfield's completeness theorem). All this work is based on the key notion of consistency property.

We introduce a fuzzy type theory and its calculus in order to model opinions. We begin revisiting a classical correspondence between type theories and display map categories, then move to the enriched setting so that we can account for different degrees of belief in a given argument. Finally, we write new rules taking into account the fuzziness, and prove completeness and validity for such a calculus. This is a work in progress and the first part of a project for the Adjoint School of 2022, it is joint with S. Arya, P. North, S. O’Connor, H. Reiss, and A. L. Tenório.

The aim of this talk is to present equational theories. Definable sets are the main objects of study in model theory. In many theories, such as algebraically closed fields and vector spaces, there is a family of sets with tame properties such that all the definable sets of the theory are a finite Boolean combination of these sets. Equationality is a generalisation of this concept. We start defining the notion of equations. Then we show that it is stronger than stability and the connection with indiscernible closure. We conclude with some examples.

Equality in First Order Logic is fairly well understood: from a syntactic point of view it can be characterised as a binary predicate forced to be reflexive and substitutive, and from a categorical perspective, following Lawvere, it can be described in terms of left adjoints. This story can be easily rephrased in the context of predicate Linear Logic, however, this smooth approach has an unexpected consequence: equality can be used an arbitrary number of times. This fact is not desirable in a substructural setting where one aims at controlling the use of resources. Moreover, it does not allow a quantitative interpretation of equality, for instance, as a distance.
In this talk, we explore a novel approach to equality in substructural logic based on graded modalities. These modalities allow us to explicitly model resources inside the language, describing how much a formula can be used. In this way, we manage to control the use of equality, thus enabling its quantitative interpretation. We develop this approach using the categorical language of Lawvere's doctrines and having as main example metric spaces with Lipschitz maps. We also present a deductive calculus for (fragments of) predicate Linear Logic with this quantitative equality and its sound and complete categorical semantics. Finally, we describe a universal construction producing models of quantitative equality starting from models of (fragments of) Linear Logic with graded modalities.
This is joint work with Fabio Pasquali presented at LICS 2022.

The first part of this talk will be a survey of the state of the art in the model theory of ordered abelian groups, with emphasis on quantifier elimination. In the second part, I will present current work in progress of myself and Jan Dobrowolski, focused on "generic" automorphisms of ordered abelian groups and of ordered real vector spaces.

In this talk we recall the relations of embeddability and convex embeddability on the set \(\mathsf{LO}\) of countable linear orders. We extend the notion of convex embeddability providing a family of quasi-orders on \(\mathsf{LO}\) of which embeddability is a particular case as well. We study these quasi-orders from a combinatorial point of view and analyse their complexity with respect to Borel reducibility, highlighting differences and analogies with embeddability and convex embeddability.

Four decades after the invention of forcing, Laver and independently Woodin answered one of the most natural questions regarding forcing. Is the ground model definable in its forcing extensions? Surprisingly, it turns out that the ground models of a given set-theoretic universe are uniformly definable. Fuchs, Hamkins and Reitz used this result to establish the formal foundations for set-theoretic geology that reverses the forcing construction by studying what remains from a model of set theory once the layers created by forcing are removed. Such a switch in perspective leads to another interesting question. Is the universe itself a nontrivial forcing extension of a smaller model? Reitz addressed the issue and introduced the Ground Axiom (the precursor to set-theoretic geology) which asserts that the universe is not obtained by forcing over any strictly smaller model.
This talk is about some types of inner models which are defined following the paradigm of “undoing” forcing. For example, a bedrock is a ground satisfying the Ground Axiom and the mantle is the intersection of all grounds. Once the main geological notions are in place, we will introduce inner models with the cover and approximation properties called pseudo-grounds. In particular, we will consider some generalizations of classical results to the context of class forcing and pseudo-grounds.

In this talk we study principles related to the (induced) subgraph problem using Weihrauch reducibility. Such problems are well studied in finite complexity theory, but they can be naturally generalized to the infinite case. After a brief introduction on computable analysis and Weihrauch reducibility, we solve some open questions in a recent article of BeMent, Hirst and Wallace. Here the authors studied the Weihrauch degrees of problems (that in this talk we denote by \(\mathsf{FindSG}_G\) and \(\mathsf{FindIndSG}_G\) respectively) that, given in input a computable graph \(H\), output \(1\) if \(G\) is an (induced) subgraph of \(H\). The authors proved that for a computable non-empty graph \(\mathsf{LPO}\leq_\mathrm{W}\mathsf{FindIndSG}_G\leq_\mathrm{W}\mathsf{WF}\), leaving open the question whether there is a graph \(G\) such that \(\mathsf{FindIndSG}_G\) lies strictly in between them. We will negatively answer this question and improve their results about the subgraph decision problems.
We then introduce strictly related principles. Such principles, given in input a computable graph \(H\) having \(G\) as an (induced) subgraph, output an isomorphic copy of \(G\). We will show how these relates with well-studied principles in the Weihrauch lattice.
This is a joint work with Arno Pauly (Swansea University).

Would set-theorists miss out on a lot if they didn't care about other methods for constructing models of Set Theory and only used forcing? In this talk I will sketch out a line of reasoning I follow in my thesis, under the supervision of Prof. Matteo Viale, in the pursuit of a more rigorous answer to a formalized aspect of the question of how powerful forcing is. The goal is to argue that a rich class of models of set theory are accessible through forcing from easily accessible standard structures — of the form \(H_\delta\).

In this seminar I will present a join work with my supervisor Marino Miculan (see arXiv:2110.10970). I'll present a formal system for fuzzy algebraic reasoning: this sequent calculus is based on two kinds of propositions, capturing equality and existence of terms as members of a fuzzy set. I'll provide a sound semantics for this calculus and show that there is a notion of free model for any theory in this system, allowing us (with some restrictions) to recover models as Eilenberg-Moore algebras for some monad. I will also prove a completeness result: a formula is derivable from a given theory if and only if it is satisfied by all models of the theory. Finally, if possible, I'll show how to use some results by Milius and Urbat to give an HSP-like characterizations of subcategories of algebras which are categories of models of particular kinds of theories.

Team Semantics was introduced by Hodges as a generalisation of the standard Tarski's semantics of first order logic. While in the usual first-order model theory free variables are interpreted via assignments, in team semantics they are interpreted via teams, i.e. set of assignments. This framework was employed by Jouko Väänänen to introduce and develop dependence logic and related formalisms — inclusion logic, independence logic, etc. — which extend first order logic by suitable atoms. In this talk, I shall introduce the underlying ideas of team semantics and focus on some open problems in the model theory of (in)dependence logic. Firstly, I will present a novel proof of the compactness of (in)dependence logic, which strengthens previous results by Kontinen, Yang and Väänänen. Secondly, I will introduce types in team semantics and dependence logic and prove some preliminary results about the topological space of types of dependence logic. This is a joint work-in-progress with Joni Puljujärvi.

In 2009 Brian Semmes, in his PhD thesis, provided a characterization of Borel measurable functions from and into the Baire space using a reduction game called the Borel game. Around the same year, Alain Louveau wrote some (still unpublished) notes in which he provided a characterization of Baire class $\alpha$ functions (again from and into the Baire space), for all fixed $\alpha$ and, importantly, $\boldsymbol{\Sigma}_\lambda^0$-measurable functions for $\lambda$ countable limit, using tree-representations instead of games. In this talk, we present Louveau's characterization, comparing it with Semmes' one, and see that if we modify a bit the Borel game we end up characterizing functions having a $G_\delta$ graph. Then we notice that under $\mathrm{AC}$ there are functions for which the Borel game is undetermined, thus opening questions regarding the consistency strength of the general determinacy of this game.

Generalized descriptive set theory (GDST) aims at developing a higher analogue of classical descriptive set theory in which is replaced with an uncountable cardinal in all definitions and relevant notions. In the literature on GDST it is often required that , a condition equivalent to regular and . In contrast, in this talk we use a more general approach and develop in a uniform way the basics of GDST for cardinals still satisfying but independently of whether they are regular or singular. This allows us to retrieve as a special case the known results for regular , but it also uncovers their analogues when is singular. We also discuss some new phenomena specifically arising in the singular context (such as the existence of two distinct yet related Borel hierarchies), and obtain some results which are new also in the setup of regular cardinals, such as the existence of unfair Borel codes for all Borel sets.

We gently review some definitions and theorems regarding the free semigroup of words, from two different areas: automata theory and Ramsey theory. Our recent results in Ramsey theory hint at a possible connection between two classes of monoids introduced by Solecki and "[the second] most important result of the algebraic theory of automata" (J. E. Pin). While this possibility is still vague, it seems exciting enough to be investigated. As far as prerequisites are concerned, most of this talk could be followed by anyone having a bachelor in mathematics. Based on a joint work with Claudio Agostini.

Wadge reducibility is a very important tool in descriptive set theory. On 0-dimensional Polish spaces it yields a nice hierarchy and very studied by results of Wadge, Martin, Monk, Andretta, Louveau, Duparc, Carroy, Medini, Müller, Motto Ros, and others. On Cantor space and Baire space the Wadge hierarchy has some different behaviors, one of those on countable degree. Namely countable degree on Cantor space are non selfdual classes, while on Baire space they are selfdual classes. This phenomenon arise a question: what happens in general 0-dimensional Polish spaces? We will answer this question with the notion of compactness degree for a 0-dimensional Polish space and see that infinitely different many cases can be realized on 0-dimensional Polish spaces.

Wadge reducibility is a very important tool in descriptive set theory. On 0-dimensional Polish spaces it yields a nice hierarchy and very studied by results of Wadge, Martin, Monk, Andretta, Louveau, Duparc, Carroy, Medini, Müller, Motto Ros, and others. On Cantor space and Baire space the Wadge hierarchy has some different behaviors, one of those on countable degree. Namely countable degree on Cantor space are non selfdual classes, while on Baire space they are selfdual classes. This phenomenon arise a question: what happens in general 0-dimensional Polish spaces? We will answer this question with the notion of compactness degree for a 0-dimensional Polish space and see that infinitely different many cases can be realized on 0-dimensional Polish spaces.

Hindman's theorem states that for any finite coloring of any semigroup you can find an infinite sequence such that the finite ordered products of this sequence are monochromatic. In this talk we will see similar theorems in the context where a monoid is acting on semigroups. In this context, you can ask whether for any finite coloring you can find a monochromatic "combinatorial span" of some sequence. There are monoids, called Ramsey monoids, for which for any semigroup you can find a combinatorial span (more precisely: for any semigroup where the monoid action can be defined). First examples were found by Carlson in 1988 and Gowers in 1992.
I will briefly state some new results obtained in a joint paper with Claudio Agostini, which is work in progress, that extend recent results from S. Solecki, who brought beautiful ideas in this context. Mainly, though, I will speak about methods, focusing on the quest for idempotents in right topological semigroups. This quest is one of the main difficulties to prove this kind of theorem. No logic background is required, since I will barely touch the part of the proofs where it is required.

Hindman's theorem states that for any finite coloring of any semigroup you can find an infinite sequence such that the finite ordered products of this sequence are monochromatic. In this talk we will see similar theorems in the context where a monoid is acting on semigroups. In this context, you can ask whether for any finite coloring you can find a monochromatic "combinatorial span" of some sequence. There are monoids, called Ramsey monoids, for which for any semigroup you can find a combinatorial span (more precisely: for any semigroup where the monoid action can be defined). First examples were found by Carlson in 1988 and Gowers in 1992.
I will briefly state some new results obtained in a joint paper with Claudio Agostini, which is work in progress, that extend recent results from S. Solecki, who brought beautiful ideas in this context. Mainly, though, I will speak about methods, focusing on the quest for idempotents in right topological semigroups. This quest is one of the main difficulties to prove this kind of theorem. No logic background is required, since I will barely touch the part of the proofs where it is required.

Model-theoretic stability is a concept having multiple facets and in fact there are at least three ways for a theory to be stable: (1) It cannot code an infinite linear order; (2) All its types are definable; (3) The size of its type space is small. The more involved implication is from (1) to (2) and all its classical proofs are combinatorial (discrete) in nature. After I. Ben Yaacov, we present an elegant new proof of the above implication in continuous logic which crucially involves a characterization result for compactness of continuous maps and briefly propose a possible interpretation. To do so, we first discuss stability in the classical case, presenting also an alternative combinatorial proof of (1) implies (3); second, we deal with compactness for continuous maps and prove the characterization theorem; third, we give an introductive account of continuous logic with a focus on a possible functional analysis interpretation; finally, we prove the equivalence of the three above conditions in continuous logic, with some final remarks.

Model-theoretic stability is a concept having multiple facets and in fact there are at least three ways for a theory to be stable: (1) It cannot code an infinite linear order; (2) All its types are definable; (3) The size of its type space is small. The more involved implication is from (1) to (2) and all its classical proofs are combinatorial (discrete) in nature. After I. Ben Yaacov, we present an elegant new proof of the above implication in continuous logic which crucially involves a characterization result for compactness of continuous maps and briefly propose a possible interpretation. To do so, we first discuss stability in the classical case, presenting also an alternative combinatorial proof of (1) implies (3); second, we deal with compactness for continuous maps and prove the characterization theorem; third, we give an introductive account of continuous logic with a focus on a possible functional analysis interpretation; finally, we prove the equivalence of the three above conditions in continuous logic, with some final remarks.

Boolean valued models provide a flexible tool to produce a variety of structures. The standard presentation introduces them via a generalization of Tarski semantics for first order logic. We show here that this approach is equivalent to a categorial approach which introduces them as a certain class of presheaves. We also outline how central concepts in the theory of boolean valued models, which are: the mixing property (used in the forcing method to produce "nice names” for certain type of sets), and the fullness property (the key tool in establishing the forcing theorem, or equivalently the Lòs theorem for boolean valued models) can be elegantly formulated in the language of sheaves and bundles. Specifically we can show that the mixing property holds for a boolean valued model if and only if its presheaf type is a sheaf. The formulation of the fullness property in the sheaf-theoretic terminology is more delicate and requires to analyze the notion of ètalè space associated to a boolean valued model. This is Joint work with Matteo Viale

Boolean valued models provide a flexible tool to produce a variety of structures. The standard presentation introduces them via a generalization of Tarski semantics for first order logic. We show here that this approach is equivalent to a categorial approach which introduces them as a certain class of presheaves. We also outline how central concepts in the theory of boolean valued models, which are: the mixing property (used in the forcing method to produce "nice names” for certain type of sets), and the fullness property (the key tool in establishing the forcing theorem, or equivalently the Lòs theorem for boolean valued models) can be elegantly formulated in the language of sheaves and bundles. Specifically we can show that the mixing property holds for a boolean valued model if and only if its presheaf type is a sheaf. The formulation of the fullness property in the sheaf-theoretic terminology is more delicate and requires to analyze the notion of ètalè space associated to a boolean valued model. This is Joint work with Matteo Viale

Martin's conjecture is a famous open problem about functions preserving Turing equivalence, whose aim was explaining why the Turing degrees of all "concrete" decision problems seem to form a well-order. A fundamental partial result was given by Slaman and Steel, who proved that Martin's conjecture holds when restricted to those functions which preserve Turing equivalence in a uniform way.
In this series of seminars, we will focus on such uniform functions, and we will first review the background on Martin's conjecture, and then we will show that part of Slaman and Steel's result arises locally, that is, can be gotten from facts concerning the internal structure of each Turing degree as provided by Turing reductions.
Using this local approach, we shall also give an easier proof of a related theorem of Lachlan and present applications both in the metamathematics of Martin's conjecture and in the theory of computable reducibilty between equivalence relations on natural numbers.

Martin's conjecture is a famous open problem about functions preserving Turing equivalence, whose aim was explaining why the Turing degrees of all "concrete" decision problems seem to form a well-order. A fundamental partial result was given by Slaman and Steel, who proved that Martin's conjecture holds when restricted to those functions which preserve Turing equivalence in a uniform way.
In this series of seminars, we will focus on such uniform functions, and we will first review the background on Martin's conjecture, and then we will show that part of Slaman and Steel's result arises locally, that is, can be gotten from facts concerning the internal structure of each Turing degree as provided by Turing reductions.
Using this local approach, we shall also give an easier proof of a related theorem of Lachlan and present applications both in the metamathematics of Martin's conjecture and in the theory of computable reducibilty between equivalence relations on natural numbers.

In this talk we analyze the the complexity of some classification problem for proper arcs and knots using Borel reducibility. We prove some anticlassification results by adapting a recent result of Kulikov on the classification of wild proper arcs/knots up to equivalnce. Moreover, we prove various new results on the complexity of the (oft-overlooked) sub-interval relation between countable linear orders, showing in particular that is at least as complicated as the isomorphism relation between linear orders.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.