#### Talks during the spring term 2019

Friday 18.1.2019 14-15, B120

Michael Harris: Automorphic Galois representations and the Langlands program

Abstract: Galois representations encode many of the classical questions of number theory. The theory of automorphic forms lies at the border between geometry and analysis on homogeneous spaces,.The Langlands program is predicated on the insight that these two theories mirror each other with astonishing accuracy. This talk will introduce some of the essential questions in the Langlands program and will indicate some recent progress.

Wed 23.1.2019 12-14, C124

no seminar (due to arctic set theory workshop)

**Extra seminar:** Monday 28.1. 10-12, B119

John T. Baldwin: Strongly minimal Steiner systems: Zilber's conjecture, universal algebra, and combinatorics , Abstract

Wed 30.1.2019 12-14, C124

Andrés Villaveces: Around the Galois group of an AEC

Abstract: I will provide a kind of blueprint for the study of interpretations of Abstract Elementary Classes:

first, I will revisit interpretability and internality in a category-theoretical language (for first order theories, reframing work of Hrushovski and Kamensky in a formalism derived from Makkai). I will then describe the issue of recovering the biintepretability class of a theory in terms of the automorphism group of a saturated model, and the role of the “Small Index Property” (SIP). An SIP theorem for AECs with strong amalgamation properties we published with Ghadernezhad in 2017 is now placed in the context of reconstruction: I propose notions of interpretation between some specific kinds of AECs and explore the role of a Galois group for an AEC.

Wed 6.2.2019 12-14, C124

Jouko Vänäänen: On Shelah's infinitary logic L-one-kappa.

Abstract: Shelah introduced some years ago a new infinitary logic L-one-kappa. We give an alternative formulation and prove that it is equivalent to the original one. Our alternative has some benefits over Shelah's version. In particular, Shelah's logic lacks - in a sense - syntax, while ours has - in a sense - syntax. This is joint work with A. Villaveces and B. Velickovic.

Wed 13.2.2019 12-14, C124

Philip Welch: Closed and Unbounded Classes and the Härtig Quantifier Model

Abstract: We investigate generalisations of the model that arise by enlarging the Gödelian operation of definability to include definability in a language with the "equicardinality quantifier", roughly that the extension of two monadic predicates has the same cardinality in the universe of all sets. Such a model is the Gödel closure of a cub (closed and unbounded) class of ordinals P, which set theorists might write as ( L[P], \in, P ). (The Härtig quantifier model is obtained by taking P = Card - a predicate true of just the cardinal numbers). Under modest strengthenings of the ZFC axioms we can obtain a full picture of what L[Card] must be like. But more than that, we may classify almost all models of the form L[P].

Wed 20.2.2019 12-14, C124

Matti Järvisalo: On the Unexpected Effectiveness of Propositional Satisfiability in Practice

Abstract: In theory, the propositional satisfiability problem (SAT) is NP-complete and no non-brute-force algorithms for it are known. In practice, SAT is a success story of modern computer science. Practical implementations of complete decision procedures for SAT, i.e., SAT solvers, act as "practical NP-oracles" in best practical algorithmic approaches to a wide range of NP and beyond-NP problems, from hardware and software verification to planning and scheduling (and beyond). In this talk, I aim to give a practically-oriented introductory overview of SAT solving, including---as time allows it---a historical perspective and modern algorithmic techniques, connections between SAT solving and propositional proof systems, and how SAT solvers can be employed for solving optimization problems and reasoning problems with (presumable) beyond-NP complexity.

Wed 27.2.2019 12-14, C124

Fausto Barbero: Causal teams

Wed 6.3.2019 12-14, C124

exam week (no seminar)

Wed 13.3.2019 12-14, C124

Pietro Galliani: Some Results on the Characterization of Safe and Strongly First Order Dependencies

Abstract: An open problem in the area of Team Semantics consists in the characterization of the dependency atoms, or of the families of dependency atoms, that are "strongly first order" in the sense that they can be added to the language of First Order Logic without increasing its expressive power (wrt sentences). A related problem consists in the characterization of the atoms S that are safe for other atoms D, in the sense that any sentence of FO(D,S) is equivalent to some sentence of FO(D); and, of course, the same questions may also be asked about operators (rather than merely dependencies) in Team Semantics. In this talk I will present a number of partial solutions to these problems, including in particular a full characterization of the downwards closed dependencies that are strongly first order and relativizable.

Wed 20.3.2019 12-14, C124

Jouko Väänänen: On Shelah's infinitary logic L-one-kappa: Part II.

Abstract: Shelah introduced some years ago a new infinitary logic L-one-kappa. We give an alternative formulation and prove that it is equivalent to the original one. Our alternative has some benefits over Shelah's version. In particular, Shelah's logic lacks - in a sense - syntax, while ours has - in a sense - syntax. This is joint work with A. Villaveces and B. Velickovic.

Wed 27.3.2019 12-14, C124

Tabea Rohr (Jena): "Frege's concept of logic reconsidered"

According to the Dreben school of Frege interpretation Frege had a universalist conception of logic. For this reason, Frege could not develop a metaperspective on logic. Thus, he does not have a wholesale, but a retail concept of logic: Logical sentences are those sentences which can be deduced from sentences which contain only topic universal vocabulary.

In this talk this picture of Frege is challenged. It is argued that 19h century logicians generally did not develop formal metalogical notions. However, Frege does distinguish between logic as such and a particular formalization of it, such as the concept script. Logic differs from other sciences because it is more general. In the talk the more-general-than relation is worked out more clearly and interpreted in the context of the development of non-Euclidean geometry in the 19^{th} century.

Wed 3.4.2019 12-14, C124

Vera Fischer (Vienna)

Title: INDEPENDENCE AND ALMOST DISJOINTNESS

Abstract: The cardinal characteristics of the real line arise from various combinatorial properties of the reals. Their study has already a long history and is closely related to the development of major forcing techniques among which template iterations (see [9]), matrix iterations (see [2], [4]) and creature forcing (see [8]). An excellent introduction to the subject can be found in [1].

Two of classical combinatorial cardinal characteristics of the continuum are the almost disjointness and independence numbers. An infinite family A of infinite subsets of N whose elements have pairwise finite intersections and which is maximal under inclusion, is called a maximal almost disjoint family. The minimal size of such a family is denoted a and is referred to as the almost disjointness number. A family A of infinite subsets of N with the property that for any two finite disjoint subfamilies F and G, the set F\G is infinite is said to be independent. The minimal size of a maximal independent family is denoted i and is referred to as the independence number. The characteristics a and i are among those for which there are no other known upper bounds apart from c, the cardinality of the continuum. It is well known that consistently a < i, however both the consistency of i < a and the inequality a ≤ i remain open.

In this talk, we will outline some of the major properties of independence and almost disjoint- ness, describe recent results (see for example [7, 5, 3, 6, 7]) and point out further interesting open problems.

Reference

[1] A. Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory., Vols. 1, 2, 3, 395–489, Springer, Dordrecht, 2010.

[2] A. Blass, S. Shelah, Ultrafilters with small generating sets, Israel Journal of Mathematics., Vols. 1, 2, 3, 395–465(3):259 – 271, 1989

[3] J. Brendle, V. Fischer, Y. Khomskii, Definable independent families, Transactions of the American mathematical Society, to appear.

[4] V. Fischer, S. Friedman, D. Mej́ıa, D. Montoya, Coherent systems of finite support iterations, Journal of Symbolic Logic, 83 (2018), no. 1, 208–236.

[5] V. Fischer, D. Mej́ıa, Splitting, bounding and almost disjointness can be quite different, Canadian Journal of Mathematics, 69 (2017), no. 3, 502 - 531.

[6] V. Fischer, D. Montoya, Ideals of independence, Archive for Mathematical Logic, to appear.

[7] V. Fischer, S. Shelah, The Spectrum of Independence, Archive for Mathematical Logic to appear, DOI:10.1007/s00153-019-00665-y;

[8] S. Shelah, On cardinal invariants of the continuum, Axiomatic set theory (Boulder, Colo., 1983)183–207, Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984.

[9] S. Shelah, Two cardinal invariants of the continuum (d < a) and FS linearly ordered iterated forcing., Acta Mathematica 192 (2004), no. 2, 187–223.

Wed 10.4.2019 12-14, C124

Raine Rönnholm: On the relationship between anonymity and dependence

Wed 17.4.2019 12-14, C124

Fan Yang: Logics for first-order team properties

Abstract: In this talk, we introduce a logic based on team semantics, called FOT, whose expressive power coincides with first-order logic both on the level of sentences and (open) formulas, and we also show that a sublogic of FOT, called FOT↓, captures exactly downward closed first-order team properties. We axiomatize completely the logic FOT, and also extend the known partial axiomatization of dependence logic to dependence logic enriched with the logical constants in FOT↓.

Wed 24.4.2019 12-14, C124

Arnaud Durand (Paris): Recursion schemes, discrete differential equations and characterization of polynomial time computation

Abstract: In this talk, we study the expressive and computational power of discrete Ordinary Differential Equations (ODEs) i.e differential equations based on the finite differences derivative. We present a framework using discrete ODEs as a central tool for computation and algorithm design.

The talk will be in two parts:

- In the first part, we will review basic results on the general theory of discrete ODE and also survey different approaches used in the literature to capture complexity classes through restricted recursion schemes.

- In the second part, we will present more specific tools and notions that permit to capture functions computable in polynomial time by discrete ODE (and if time permits, NP and PSPACE).

Wed 1.5.2019 12-14, C124

Vappu, no seminar

Wed 8.5.2019 12-14, C124

exam week