TBA

### Past:

• 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:

• 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".