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Talks during the spring term 2018

Wed 17.1.2018 12-14, C124
Yurii Khomskii: Projective maximal independent families

Wed 24.1.2018 12-14, C124
Fan Yang: Deriving and generalizing Arrow’s Theorem in dependence and independence logic

Wed 31.1.2018 12-14, C124
Fan Yang: Deriving and generalizing Arrow’s Theorem in dependence and independence logic, continued

Wed 7.2.2018 12-14, C124
Tim Gendron: Quasicrystals and the Quantum Modular Invariant

Abstract: Hilbert’s 12th problem asks for an explicit description of the Hilbert and ray class fields of a global field $K$. It was inspired by the Theorem of Weber-Fueter, where it is shown that the Hilbert class field of $K$ quadratic and complex over $\mathbb{Q}$ is generated by the modular invariant of any element of $K-\mathbb{Q}$. For $K=\mathbb{Q}(\theta )$ a real quadratic extension of $\mathbb{Q}$ we conjecture that the Hilbert class field is generated by a weighted product of the values of $j^{\rm qt}(\theta)$ where $j^{\rm qt}$ is a discontinuous and multi valued function called the quantum modular invariant. In the case of a real quadratic field of positive characteristic, this conjecture has been verified using the theory of Drinfeld-Hayes modules, wherein the values of $j^{\rm qt}$ are shown to be the modular in invariants of certain ideals in a “small” Dedekind ring. Recently Richard Pink has shown that the same is true in characteristic zero using a quasicrystalline analog of Dedekind ring. We show that the set of quasicrystalline ideals naturally form a Cantor set on which the modular invariant is continuous. We end by considering the new frontier of quasicrystalline algebraic number theory and its prospects for providing a basis for a Drinfeld-Hayes theory in characteristic zero that may eventually allow one to solve Hilbert’s 12th problem for real quadratic extensions of $\mathbb{Q}$.

Wed 14.2.2018 12-14, C124
Fan Yang: Questions and dependency in intuitionistic logic

Wed 21.2.2018 12-14, C124
Fausto Barbero: Interventionist counterfactuals in causal team semantics

Abstract: Teams and multiteams are adequate semantical objects for the discussion of properties of data, such as database dependencies or probabilities. There are instead notions of dependence - such as the causal dependencies arising from manipulationist theories of causation            (Pearl, Woodward) - which cannot be reduced to properties of data. These sorts of dependencies are meaningful in presence of a set of basic causal assumptions - a set of counterfactual statements, which are usually summarized by so-called structural equations.    However, theories of causation make a mixed use of observational and probabilistic notions (which concern data) and of causal notions. I will show how all these forms of reasoning can be modeled within one single semantical framework which simultaneously extends team semantics and structural equation models; and I will analyze some aspects of the logic of interventionist counterfactuals that emerges from this approach. (Joint work with Gabriel Sandu)

Wed 28.2.2018 12-14, C124
Miguel Moreno: \Sigma_1^1-complete quasi-orders in L cancelled due to strike

Wed 7.3.2018 12-14, C124
exam week

Wed 14.3.2018 12-14, C124
Miikka Vilander: A formula size game for the modal mu-calculus

Wed 21.3.2018 12-14, C124
Miguel Moreno: \Sigma_1^1-complete quasi-orders in L

Abstract: One of the basic differences between descriptive set theory (DST) and generalized descriptive set theory (GDST) is the existence of some analytic sets in GDST that have no counterpart in DST. The equivalence modulo the non-stationary ideal restricted to a stationary set S (EM-S) is an example of these sets.

These relations have been studied in GDST to understand some of the differences between DST and GDST. In particular, EM-S, with S the set of ordinals with cofinality alpha, has been used to study the isomorphism relations in the Borel-reducibility hierarchy. One of the main results related to this is:

    • If V=L, then EM-S, with the set of ordinals with cofinality alpha, is a complete analytic equivalence relation in the generalized Baire space.

From this, it was possible to prove that the isomorphism relation of theories with OCP or S-DOP are complete analytic equivalence relations in L(under some cardinal assumptions).

In this talk I will show that the inclusion modulo the non-stationary ideal restricted to the set of ordinals with cofinality alphais is a complete analytic quasi-order in L. This result has many corollary, two of which are:

    • (V=L) The isomorphism relation of a theory is either a complete analytic equivalence relation or a Delta_1^1  equivalence relation (under some cardinal assumptions).
    • (V=L) If kappa is not the successor of an omega-cofinal cardinal, then the embedability of dense linear orders is a complete analytic relation.

Wed 28.3.2018 12-14, C124
no seminar

Wed 4.4.2018 12-14, C124
Jouko Väänänen: “Internal categoricity of first order set theory”

Abstract: We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership-relation is E and also when the membership-relation is E', and in both cases the formulas in schemata are allowed to contain both E and E', then (M,E) and  (M,E’) are isomorphic. We offer this as a strong indication of 'internal categoricity' of set theory.

Wed 11.4.2018 12-14, C124
Oystein Linnebo: "Generality explained" 

Abstract: What explains a true universal generalization? I distinguish and investigate different kinds of explanation. While an instance-based explanation proceeds via (all or some) instances of the generalization, a generic explanation is independent of the instances, relying instead on completely general facts about the properties or operations involved in the generalization. This distinction is illuminated by means of a truthmaker semantics, which is also used to show that instance-based explanations support classical logic, while generic explanations support only support intuitionistic logic.

Wed 18.4.2018 12-14, C124
Maria Hämeen-Anttila : "Gödel's early views on intuitionism"

Abstract: In the early 1930s, Kurt Gödel made several contributions to intuitionistic logic. He then turned to the question of constructivity of intuitionistic logic. His early critique of intuitionism is known from three lectures given in 1933, 1938, and 1941, where he claimed that the proof interpretation, as well as the intuitionistic treatment of negated universal statements, is not strictly constructive. I examine Gödel's criticism in light of his published works and unpublished notes, especially those from Gödel's 1941 lecture course in Princeton. I will also suggest that his view of intuitionism and the confusion about intuitionistic quantifiers came from, not Brouwer, whose works Gödel was not acquainted with at the time, but rather Weyl's and Hilbert's interpretation of Brouwer's intuitionism.

Wed 25.4.2018 12-14, C124
Nemi Pelgrom: Introduction to alternative foundations - univalence and type theory.

Abstract: With the rise of interest in formalisation of mathematics in the past century, and the rise of interest in proof assistants in the last decades, there are some alternative foundations to mathematics that are being tried out. Constructive mathematics and type theory has been shown to be preferable, over classic set theory, in programming contexts. This was first shown by H. Curry and W. A. Howard with work in the middle of the 20th century now know as “the Curry-Howard correspondence”, and has recently reached new levels in the emerging field called Homotopy type theory. The Univalence axiom introduced by V. Voevodsky just a couple of years ago, allows us to treat “identity as equivalent to equivalence”, and this is central to the success of HoTT. This talk will introduce some necessary background for understanding what this axiom means and what HoTT is, and also discuss this theory’s current and future place in the foundation of mathematics. The talk with be of a philosophical and historical rather than mathematical nature.

Wed 2.5.2018 12-14, C124

Wed 9.5.2018 12-14, C124
Philip Welch: Recursions of higher types and low levels of determinacy

Abstract: We explore how generalisations of Kleene's theory of recursion in type 2 objects (which can be used to characterise complete Pi^1_1 sets and open =  Sigma^0_1-determinacy) can be lifted to Sigma^0_3-Determinacy. The latter is the highest level in the arithmetical hierarchy whose determinacy is still provable in analysis. The generalisation requires the use of so-called infinite time Turing machines, and the levels of the Gödel constructible hierarchy needed to see that such machines models produce an output are, perhaps surprisingly, intimately connected with those needed to prove the existence of such strategies. The subsystem of analysis needed for this work is Pi^1_3-CA_0, and there is something suggestive of what may be needed to give a proof theory for this latter theory.

Short courses and lecture series in the spring 2018:

May 14-17, C122, 14-16: Advanced course "Forcing, forcing axioms and a hope for having them at cardinals other than aleph_1" by Mirna Dzamonja

May 22-24, C122/C124, 11-12: Minicourse on computable analysis by Arno Pauly

Menachem Magidor: May 22, during the hours of 14-16, C122, Exactum Building, Kumpula


Abstract: In this talk we shall survey some results about compactness and reflection for some second order properties of mathematical structures.

 LINK: Young Set Theory Lecture 1 handout.pdfmagidor2018.pdf

Menachem Magidor: May 25, during the hours of 14-16, C124, Exactum Building, Kumpula

Title: INDEPENDENCE IN MATHEMATICS, IS IT RELEVANT? LINK:  Helsinki Logic Group May 2018.pdf

Abstract:  Gödel’s incompleteness theorem states that in any mathematical theory, except for trivial theories, there are statements that can not decided on the basis of the theory. Namely, these statements are independent of the theory. While Gödel’s theorem has a deep impact on Logic and the Philosophy of Mathematics, initially it got relatively small attention from the general mathematical public. The feeling was that the Gödelian statements are ”artificial ” and do not come up in the usual mathematical practice.

The situation was changed to a large extent with the discovery that central problems of SetTheory, like the Continuum Hypothesis, are independent of the usual axioms of Set Theory. It was followed by proofs that many other problems in different areas of Mathematics are independent of the usual axioms.

In spite of that, several authors claimed that these independence results have no effect on the core practice of Mathematics---that problems which are considered to be in the mainstream of mathematical activity are not independent. The problem is even more pertinent for the applications of Mathematics in science. Namely: Does the phenomenon of independence in Mathematics have any relevance in the use of mathematical models in Science?

While we do not have concrete examples of a scientific theory, (say in theoretical physics) that is sensitive to the truth of a mathematical statement which is independent, we shall try to argue that it is not impossible. The issue raises some basic questions about the application of Mathematics in Science.

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