## Turin-Udine logic seminar

All seminars will be held remotely using Webex. Please write to luca.mottoros [at] unito.it to obtain a link and the access code.

TBA

### Past:

• June 18th, 2021, 16.30-18.30 (Online on WebEx)

C. Brech (Universidade de São Paulo) "Isomorphic combinatorial families" (Video).

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.

• June 11th, 2021, 16.30-18.30 (Online on WebEx)

V. Gitman (CUNY Graduate Center) "The old and the new of virtual large cardinals" (Video).

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 $\omega$-Erd\H os 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.

• June 4th, 2021, 16.30-18.30 (Online on WebEx)

M. Pinsker (Vienna University of Technology) "Uniqueness of Polish topologies on endomorphism monoids of countably categorical structures" (Video).

"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."

• May 28th, 2021, 16.30-18.30 (Online on WebEx)

D. Bartosova (University of Florida) "Short exact sequences and universal minimal flows" (Video).

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.

• May 21st, 2021, 16.30-18.30 (Online on WebEx)

L. Westrick (Penn State University) "Borel combinatorics fail in HYP" (Video).

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^{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.

• May 14th, 2021, 16.30-18.30 (Online on WebEx)

R. Sklinos (Stevens Institute of Technology) "Fields interpretable in the free group" (Video).

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.

• May 7th, 2021, 16.30-18.30 (Online on WebEx)

M. Valenti (University of Udine) "Uniform reducibility and descending sequences through ill-founded orders" (Video).

"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. "

• April 30th, 2021, 16.30-18.30 (Online on WebEx)

S. Barbina (Open University) "The theory of the universal-homogeneous Steiner triple system" (Video).

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.

• April 23rd, 2021, 16.30-18.30 (Online on WebEx)

F. Loregian (Tallinn University of Technology) "Functorial Semantics for Partial Theories" (Video).

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.

• April 16th, 2021, 16.30-18.30 (Online on WebEx)

A. Poveda (Hebrew University of Jerusalem) "Forcing iterations around singulars cardinals and an application to stationary reflection" (Video).

"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."

• April 9th, 2021, 16.30-18.30 (Online on WebEx)

A. Berarducci (University of Pisa) "Asymptotic analysis of Skolem's exponential functions" (Video).

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).

• March 26th, 2021, 16.30-18.30 (Online on WebEx)

V. Dimonte (University of Udine) "The role of Prikry forcing in generalized Descriptive Set Theory" (Video).

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.

• March 19th, 2021, 16.30-18.30 (Online on WebEx)

G. Paolini (Turin) "Torsion-Free Abelian Groups are Borel Complete" (Video).

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.

• March 12th, 2021, 16.30-18.30 (Online on WebEx)

C. Conley (Carnegie Mellon University) "Dividing the sphere by rotations" (Video).

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.

• March 5th, 2021, 16.30-18.30 (Online on WebEx)

N. de Rancourt (University of Wien) "A dichotomy for countable unions of smooth Borel equivalence relations" (Video).

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.

• February 26th, 2021, 16.30-18.30 (Online on WebEx)

S. Barbina (Open University) "The theory of the universal-homogeneous Steiner triple system" (CANCELED).

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.

• February 19th, 2021, 16.30-18.30 (Online on WebEx)

P. Shafer (University of Leeds) "An inside-outside Ramsey theorem in the Weihrauch degrees" (Video).

"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."

• February 12th, 2021, 16.30-18.30 (Online On WebEx)

A. Kwiatkowska (University of Münster) "The automorphism group of the random poset".

"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."

• February 5th, 2021, 16.30-18.30 (Online on WebEx)

M. Viale (University of Turin) "Tameness for set theory" (Video).

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.

• January 29th, 2021, 16.30-18.30 (Online on WebEx)

V. Fischer (University of Wien) "The spectrum of independence" (Video).

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.

• January 22nd, 2021, 16.30-18.30 (Online on WebEx)

R. Schindler (University of Muenster) "Martin's Maximum^++ implies the P_max axiom (*)" (Video).

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.""

• January 15th, 2021, 16.30-18.30 (Online on WebEx)

A. Freund (TU Darmstadt) "Ackermann, Goodstein, and infinite sets" (Video).

This seminar is part of the event World Logic Day 2021

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.

• January 8th, 2021, 16.30-18.30 (Online on WebEx)

F. Calderoni (University of Illinois at Chicago) "The Borel structure on the space of left-orderings" (Video).

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.

• December 18th, 2020, 16.30-18.30 (Online on WebEx)

M. Eskew (Vienna) "Weak square from weak presaturation" (Video).

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.

• December 11th, 2020, 16.30-18.30 (Online on WebEx)

A. Shani (Harvard University) "Anti-classification results for countable Archimedean groups" (Video).

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.

• December 4th, 2020, 16.30-18.30 (Online on WebEx)

L. San Mauro (Vienna) "Revisiting the complexity of word problems " (Video).

"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."

• November 27th, 2020, 16.30-18.30 (Online on WebEx)

M. Viale (Turin) "Tameness for set theory" (CANCELED).

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.

• November 20th, 2020, 16.30-18.30 (Online on WebEx)

P. Holy (Udine) "Large Cardinal Operators".

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.

• November 13th, 2020, 16.30-18.30 (Online on WebEx)

C. Agostini (Turin) "Large cardinals, elementary embeddings and "generalized" ultrafilters", part 2 (Slides).

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.

• November 6th, 2020, 16.30-18.30 (Online on WebEx)

C. Agostini (Turin) "Large cardinals, elementary embeddings and "generalized" ultrafilters", part 1 (Slides).

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.

• May 29th, 2020, 16.30-17.30 (Online)

P. Holy (Udine) "Generalized topologies on 2^kappa, Silver forcing, and the diamond principle".

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.

• May 8th, 2020, 16.30-17.30 (Online)

M. Fiori Carones (Monaco) "Unique orderability of infinite interval graphs and reverse mathematics".

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)

• April 24th, 2020, 16.00-18.00 (Online)

L. Carlucci (Rome) "Questions and results about the strength of(variants of) Hindman's Theorem".

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.

• April 1st, 2020, 14.30-16.30 (on-line)

P. Holy (Udine) "Ideal Topologies".

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).

• March 13th, 2020, 14.00-16.00 (DMIF, Sala Riunioni)

L. Carlucci (Rome) "Questions and results about the strength of (variants of) Hindman's Theorem" (CANCELED).

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.

• March 3rd, 2020, 14.00-16.00 (DMIF, Sala Riunioni)

P. Holy (Udine) "Ideal Topologies" (CANCELED).

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).

• June 7th, 2019, 14.00-16.00 (DMIF, Sala Riunioni)

V. Torres-Peréz (Vienna) "Compactness Principles and Forcing Axioms without Martin's Axiom ".

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 $\mathrm{MA}_{\aleph_1}$.

• January 29th, 2019, 17.00-18.30 (DMIF, Sala Riunioni)

X. Shi (Beijing) "Large cardinals and generalized degree structures".

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.

• September 26th, 2018, 16.30-18.00 (DMIF, Aula multimediale)

P. Shafer (Leeds) "Describing the complexity of the "problem B is harder than problem A relation"".

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.

• March 20th, 2018, 14.30-16.00 (DMIF, Sala Riunioni)

W. Gomaa (Alexandria) "On the extension of computable real functions".

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.

• November 21st, 2017, 17.00-18.30 (DMIF, Aula multimediale)

V. Brattka (Munich) "How can one sort mathematical theorems".

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.

• November 16th, 2017, 14.30-16.00 (DMIF, Sala Riunioni)

R. Cutolo (Napoli) "Berkeley Cardinals and the search for V ".

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.

## Working seminar

TBA

### Past:

• April 9th, 2021, 11.00-13.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 9.

TBA

• March 26th, 2021, 11.00-13.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 8.

TBA

• March 19th, 2021, 11.00-13.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 7.

Uniformization and Basis results

• March 5th, 2021, 11.00-13.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 6.

Scales and uniformization for Pi_1^1 sets

• February 25th, 2021, 09.00-11.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 5.

Some consequences of the Parametrization Theorem (2)

• February 18th, 2021, 09.00-11.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 4.

Some consequences of the Parametrization Theorem (1)

• February 11th, 2021, 09.00-11.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 3.

Universal sets and Parametrization Theorems for Delta_1^1

• February 4th, 2021, 09.00-11.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 2.

Norms and (easy) uniformization for Pi_1^1 sets

• January 28th, 2021, 09.00-11.00 (Online)

V. Cipriani (Udine) "Reading seminar on Effective Descriptive Set Theory", part 1.

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

• May 22nd, 2020, 16.30-18.30 (Online)

M. Valenti (Udine) "The complexity of closed Salem 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.

• May 15th, 2020, 16.30-17.30 (Online)

M. Iannella (Udine) "The G_0 dichotomy".

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.

• April 3rd, 2020, 14.30-16.30 (Online)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 7.

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.

• March 27th, 2020, 14.30-16.30 (on-line)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 6.

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.

• March 20th, 2020, 14.30-16.30 (on-line)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 5.

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.

• March 20th, 2020, 11.00-12.00 (on-line)

S. Tamburlini (Udine) "Reverse mathematics of Second Order set theory".

• March 13th, 2020, 14.30-16.30 (on-line)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 4.

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.

• February 21st, 2020, 14.00-16.00 (DMIF, Aula multimediale)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 3.

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.

• February 14th, 2020, 14.00-16.00 (DMIF, Sala Riunioni)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 2.

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.

• February 6th, 2020, 14.30-16.30 (DMIF, Sala Riunioni)

M. Valenti (Udine) "Finding descending sequences in an ill-founded linear order", part 1.

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.

• December 10th, 2019, 16.30-18.00 (DMIF, Sala Riunioni)

M. Fiori Carones (Udine) "An On-Line Algorithm for Reorientation of Graphs".

• November 27th, 2019, 08.30-10.00 (DMIF, Sala Riunioni)

M. Iannella (Udine) "On the classification of wild Proper Arcs and Knots ", part 2.

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.

• November 20th, 2019, 14.30-16.30 (DMIF, Sala Riunioni)

M. Iannella (Udine) "On the classification of wild Proper Arcs and Knots ", part 1.

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.

• October 12th, 2018, 14.00-15.30 (DMIF, Sala Riunioni)

E. Lena (Udine) "I sistemi assiomatici di Tarski per la geometria".

• June 28th, 2017, 10.00-11.30 (DMIF, Aula multimediale)

M. Fiori Carones (Udine) "Espressività -in termini di classi di complessita' "catturate"- dei linguaggi logici per la rappresentazione della conoscenza".