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.

Enactivist cognitive scientists argue against symbolic representation as a fundamental principle behind perception, semantics, and cognition in general. Here we build a mathematical model of emergence of structural representations in a brain of an interacting agent which is inline with enactivism. The main result is that under some very weak assumptions about the sensory feedback and on the robot's reward function, the robot will (surprisingly) end up reconstructing an isomorphic copy of its own environmental state space. A robot arm will be considered as a concrete example. We also point out that this framework is not limited to robotics, but is expected to generate hypotheses and explanations in any study of interactive intelligence, from biology to cognitive science and artificial intelligence. The mathematical tool we use is known as semiautomaton. However, continuous/topological versions of the main results are on their way. (Partial joint work with Steven LaValle and Basak Sakcak.)

The Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras. The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this talk we introduce a weaker concept which we call the “fuzzy filter”. Using fuzzy filters instead of ultrafilters enables one to extend the class of spaces under consideration from totally disconnected ones to a larger class. As an application of this method, we solve anumber of open problems. We show that the following are completely classifiable by countable structures: the homeomorphism on 3-manifolds (also applicable to 2-manifolds; but this was known since 1971), wild embeddings of Cantor sets in R³. By classification in this talk we mean the classical Borel-reducibility (joint work with Martina Iannella).

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.

The common base theory of reverse mathematics is the theory RCA0, which guarantees the existence of Delta01 -definable sets and where mathematical induction for Sigma01 -formulae holds. In 1986, Simpson and Smith introduced a different base theory, RCA0*, where induction is weakened to Delta01 -formulae. In more recent years Kolodziejczyk, Kowalik, Wong, Yokoyama started wondering about the strength of Ramsey’s theorem over RCA0*. In this talk we concentrate on two well known consequences of Ramsey’s theorem for pairs RT22, namely the Cohesive Ramsey theorem for pairs CRT22 and the Cohesion principle COH. Over RCA0 both follow from RT22, while we proved that only CRT22, and not COH, follows from RT22 over RCA0*.

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.

I will talk about the connections between topologies on 2^kappa induced by ideals on kappa and topologies on 2^kappa induced by certain tree forcing notions, highlighting the connection of the topology induced by the nonstationary ideal with kappa-Silver forcing. Assuming that Jensen's diamond principle holds at kappa, we then generalize results on kappa-Silver forcing of Friedman, Khomskii and Kulikov that were originally shown for inaccessible kappa: In particular, I will present a proof that also in our situation, kappa-Silver forcing satisfies a strong form of Axiom A. By a result of Friedman, Khomskii and Kulikov, this implies that meager sets are nowhere dense in the nonstationary topology. If time allows, I will also sketch a proof of the consistency of the statement that every Delta^1_1 set (in the standard bounded topology on 2^kappa) has the Baire property in the nonstationary topology, again assuming the diamond principle to hold at kappa (rather than its inaccessibility). This is joint work with Marlene Koelbing, Philipp Schlicht and Wolfgang Wohofsky.

Interval graphs are graphs whose vertices can be mapped to intervals of a linear order in such a way that intervals associated to adjacent vertices have non empty intersection. For each interval graph there exists an order whose incomparability relation corresponds to the adjacency relation of the graph. In general different orders can be associated to an interval graph. We are interested to capture the class of interval graphs which have a unique, up to duality, order associated to them. In particular, we prove that a characterisation known to hold for finite connected interval graphs holds for infinite connected interval graphs as well. Finally, we settled the strength of this characterisation in the hierarchy of subsystems of second order arithmetic. (Joint work with Alberto Marcone)

Measuring the logical and computational strength of Hindman's Finite Sums Theorem is one of the main open problems in Reverse Mathematics since the seminal work of Blass, Hirst and Simpson from the late Eighties. Hindman's Theorem states that any finite coloring of the positive integers admits an infinite set such that all non-empty finite sums of distinct elements from that set have the same color. The strength of Hindman's Theorem is known to lie between ACA_0 and ACA_0^+ or, in computability-theoretic terms, between the Halting Problem and the degree of unsolvability of first-order arithmetic. In recent years, following a suggestion of Blass, researchers have investigated variants of Hindman's Theorem in which the type of sums that are required to be monochromatic is restricted by some natural constraint. Restrictions on the number of terms have received particular attention, but other structural constraints give rise to interesting phenomena as well. The main open problem remains that of finding a non-trivial upper bound on Hindman's Theorem for sums of at most two elements. By work of Kolodziejczyk, Lepore, Zdanowski and the author, this theorem implies ACA_0, yet it is an open problem in Combinatorics (Hindman, Leader, Strauss 2003) whether a proof of it exists that does not also prove the full version of the theorem. We will review some of the results from the recent past on this and related questions and discuss a number of old and new open problems.

The classical Cantor space is the space of all functions from the natural numbers to {0,1} with the topology induced by the bounded ideal -- that is the topology that is given by the basic clopen sets [f] for functions f from a bounded subset of the natural numbers to {0,1}. When generalizing this from the natural numbers to the so-called higher Cantor space on some regular and uncountable cardinal kappa, this is usually done by working with the bounded ideal on kappa. However, there are other natural ideals on such cardinals apart from the bounded ideal -- in particular there is always the nonstationary ideal on kappa. In this talk, I will define and investigate some basic properties of spaces on regular and uncountable cardinals kappa using topologies based on general ideals, and in particular on the nonstationary ideal on kappa. This is joint work with Marlene Koelbing (Vienna), Philipp Schlicht (Bristol) and Wolfgang Wohofsky (Vienna).

Measuring the logical and computational strength of Hindman's Finite Sums Theorem is one of the main open problems in Reverse Mathematics since the seminal work of Blass, Hirst and Simpson from the late Eighties. Hindman's Theorem states that any finite coloring of the positive integers admits an infinite set such that all non-empty finite sums of distinct elements from that set have the same color. The strength of Hindman's Theorem is known to lie between ACA_0 and ACA_0^+ or, in computability-theoretic terms, between the Halting Problem and the degree of unsolvability of first-order arithmetic. In recent years, following a suggestion of Blass, researchers have investigated variants of Hindman's Theorem in which the type of sums that are required to be monochromatic is restricted by some natural constraint. Restrictions on the number of terms have received particular attention, but other structural constraints give rise to interesting phenomena as well. The main open problem remains that of finding a non-trivial upper bound on Hindman's Theorem for sums of at most two elements. By work of Kolodziejczyk, Lepore, Zdanowski and the author, this theorem implies ACA_0, yet it is an open problem in Combinatorics (Hindman, Leader, Strauss 2003) whether a proof of it exists that does not also prove the full version of the theorem. We will review some of the results from the recent past on this and related questions and discuss a number of old and new open problems.

The classical Cantor space is the space of all functions from the natural numbers to {0,1} with the topology induced by the bounded ideal -- that is the topology that is given by the basic clopen sets [f] for functions f from a bounded subset of the natural numbers to {0,1}. When generalizing this from the natural numbers to the so-called higher Cantor space on some regular and uncountable cardinal kappa, this is usually done by working with the bounded ideal on kappa. However, there are other natural ideals on such cardinals apart from the bounded ideal -- in particular there is always the nonstationary ideal on kappa. In this talk, I will define and investigate some basic properties of spaces on regular and uncountable cardinals kappa using topologies based on general ideals, and in particular on the nonstationary ideal on kappa. This is joint work with Marlene Koelbing (Vienna), Philipp Schlicht (Bristol) and Wolfgang Wohofsky (Vienna).

Rado's Conjecture (RC) in the formulation of Todorcevic is the statement that every tree $T$ that is not decomposable into countably many antichains contains a subtree of cardinality $\aleph_1$ with the same property. Todorcevic has shown the consistency of this statement relative to the consistency of the existence of a strongly compact cardinal. RC implies the Singular Cardinal Hypothesis, a strong form of Chang's Conjecture, the continuum is at most $\aleph_2$, the negation of $\Box(\theta)$ for every regular $\theta\geq\omega_2$, Fuchino's Fodor-type Reflection Principle, etc. These implications are very similar to the ones obtained from traditional forcing axioms such as MM or PFA. However, RC is incompatible even with $\mathrm{MA}_{\aleph_1}$. In this talk we will take the opportunity to give an overview of our results different coauthors obtained in the last few years together with recent ones. These new implications seem to continue suggesting that RC is a good alternative to forcing axioms. We will discuss to which extent this may hold true and where we can find some limitations. We will end the talk with some open problems and possible new directions. For example, we will also discuss some recent results with Liuzhen Wu and David Chodounsky regarding squares of the form $\Box(\theta, \lambda)$ and YPFA. This forcing axiom, a consequence of PFA, was introduced by David Chodounsky and Jindrich Zapletal, where they proved that it has similar consequences as PFA, such as the P-Ideal Dichotomy, $2^{\aleph_0}= \aleph_2$, all $\aleph_2$-Aronszajn trees are special, etc. However, YPFA is consistent with the negation of $\mathrm{MA}_{\aleph_1}$.

A central task of modern set theory is to study various extensions of ZF/ZFC, to some is to search for the “right” extension of the current foundation. Large cardinal axioms were proposed by G\"{o}del as candidates, originally to settle the continuum problem. It turns out that they serve nicely as scale for measuring the strength of most "natural" statements in set theory. Recursion theory is one of the big four branches of mathematical logic. Classical recursion theory studies the structure of Turing degrees. It has been extended/generalized to higher levels of computability/definability, as well as to higher ordinals/cardinals. However these results do not go beyond ZFC. Recent studies reveal that there are deep connections between the strength of large cardinals and the complexity of generalized degree structures. I will present the latest developments in this new research program -- higher degree theory.

Some mathematical problems are harder than others. Using concepts from computability theory, we formalize the "problem B is harder than problem A" relation and analyze its complexity. Our results express that this "harder than" relation is, in a certain sense, as complicated as possible, even when restricted to several special classes of mathematical problems.

We investigate interrelationships among different notions from mathematical analysis, effective topology, and classical computability theory. Our main object of study is the class of computable functions defined over an interval with the boundary being a left-c.e. real number. We investigate necessary and sufficient conditions under which such functions can be computably extended. It turns out that this depends on the behavior of the function near the boundary as well as on the class of left-c.e. real numbers to which the boundary belongs, that is, how it can be constructed. Of particular interest a class of functions is investigated: sawtooth functions constructed from computable enumerations of c.e. sets.

It is common mathematical practice to say that one theorem implies another one. For instance, it is mathematical folklore that the Baire Category Theorem implies the Closed Graph Theorem and Banach's Inverse Mapping Theorem. However, after a bit of reflection it becomes clear that this notion of implication cannot be the usual logical implication that we teach to our undergraduate students, since all true theorems are logically equivalent to each other. What is actually meant by implication in this informal sense is rather something such as "one theorem is easily derivable from another one". However, what does "easily derivable" mean exactly? We present a survey on a recent computational approach to metamathematics that provides a formal definition of what "easily derivable" could mean. This approach borrows ideas from theoretical computer science, in particular the notion of a reducibility. The basic idea is that Theorem A is easily derivable from Theorem B if A is reducible to B in the sense that the input and output data of these theorems can be transferred into each other. In this case the task of Theorem A can be reduced to the task of Theorem B. Such reductions should at least be continuous and they are typically considered to be computable, which means that they can be performed algorithmically.The resulting structure is a lattice that allows one to sort mathematical theorems according to their computational content and phenomenologically, the emerging picture is very much in line with how mathematicians actually use the notion of implication in their daily practice.

The talk will focus on Berkeley cardinals, the strongest known large cardinal axioms, recently introduced by J. Bagaria, P. Koellner and W. H. Woodin. Berkeley cardinals are inconsistent with the Axiom of Choice; their definition is indeed formulated in the context of ZF (Zermelo-Fraenkel set theory without AC). Aim of the talk is to provide an account of their main features and the foundational issues involved. A noteworthy contribution to the topic is my result establishing the independence from ZF of the cofinality of the least Berkeley cardinal, which is in fact connected with the failure of AC; I will describe the forcing notion employed and give a sketch of the proof. In order to show that interesting mathematical consequences can be developed from Berkeley cardinals, I’ll then analyze the structural properties of the inner model $L(V_{delta+1})$ under the assumption that delta is a singular limit of Berkeley cardinals each of which is itself limit of extendible cardinals, lifting some of the theory of the large cardinal axiom I0 to a more general level. Finally, I will discuss the role of Berkeley cardinals within Woodin’s ultimate project of attaining a “definitive” description of the universe of set theory.

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.

TBA

TBA

Uniformization and Basis results

Scales and uniformization for Pi_1^1 sets

Some consequences of the Parametrization Theorem (2)

Some consequences of the Parametrization Theorem (1)

Universal sets and Parametrization Theorems for Delta_1^1

Norms and (easy) uniformization for Pi_1^1 sets

Basic notions of EDST and Representation Theorem for Pi_1^1 sets

A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. In this talk we will study the descriptive complexity of the family of closed Salem subsets of [0,1], [0,1]^n and of the n-dimensional Euclidean space.

Dichotomy theorems have always played a fundamental role in set theory, and for many decades the way to prove them was akin to the proof of Cantor-Bendixson theorem, i.e., derivative arguments. This changed in the early 1970s, when Silver proved the Silver dichotomy using sophisticated techniques borrowed from the theory of forcing and from effective descriptive set theory, ushering in a new era of dichotomy proofs. Around ten years ago, Ben Miller took upon himself to reverse back this development and to find proofs of these new results that do not rely on forcing and effective arguments, but just the good old derivative ones. The key for this is switching from equivalence relations to graphs The result of this research is a handful of dichotomies at the core of descriptive set theory that prove many other ones, with classical "easy" proofs, the most important of them being the G_0 dichotomy.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

We will explore the computational content of the problem "given an ill-founded linear order, find an infinite descending sequence" from the point of view of Weihrauch reducibility. The talk will cover the background notions on Weihrauch reducibility. The results are joint work with Jun Le Goh and Arno Pauly.

In this talk we analyze the complexity of some classification problems 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 equivalence. Moreover, we prove various new results on the complexity of the (oft-overlooked) sub-interval relation between countable linear orders, showing in particular that it is at least as complicated as the isomorphism relation between linear orders.

In this talk we analyze the complexity of some classification problems 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 equivalence. Moreover, we prove various new results on the complexity of the (oft-overlooked) sub-interval relation between countable linear orders, showing in particular that it is at least as complicated as the isomorphism relation between linear orders.