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Talks during the fall term 2020

Wed 02.09.2020, 12-14 by ZOOM
Ulla Karhumäki (University of Helsinki): Small infinite simple groups of finite Morley rank with a tight automorphism whose fixed point subgroup is pseudofinite.

We explain in detail a recent approach, developed by Pınar Uǧurlu, towards the well-known Cherlin-Zilber Conjecture. The main aim of this approach is to prove that the Cherlin-Zilber Conjecture is equivalent to another conjecture called the Principal Conjecture, which is due to Ehud Hrushovski; Let $G$ be an infinite simple group of finite Morley rank with a generic automorphism $\alpha$. Then the fixed point subgroup $C_G(\alpha)$ is pseudofinite. We prove a result supporting the expected equivalence between these two conjectures. Namely, we prove that, under suitable assumptions, a “small” infinite simple group of finite Morley rank $G$ admitting a tight automorphism $\alpha$ whose fixed point subgroup $C_G(\alpha)$ is pseudofinite is isomorphic to $PSL_2(K)$ over an algebraically closed field $K$ of positive characteristic different from 2. This is joint work with Pınar Uǧurlu.

Wed 9.9.2020
Dima Sinapova (University of Illinois at Chicago) 16:00-17:00 in Zoom
Iteration, reflection and Prikry forcing

There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. 

In contrast, failure of SCH is an instance of incompactness.  It is usually obtained using Prikry forcing. 

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we discuss the situation at smaller cardinals. This is joint work with Alejandro Poveda and Assaf Rinot.

Wed 16.9.2020, 18:00-19:00 in Zoom (Meeting ID: 638 8055 9261 Passcode: 924004 )
Phokion Kolaitis (UC Santa Cruz and IBM Research - Almaden)
The Query Containment Problem: Set Semantics vs. Bag Semantics

Query containment is a fundamental algorithmic task in database query processing and optimization. Under set semantics, the query-containment problem for conjunctive queries has long been known to be NP-complete.  SQL queries, however, are typically evaluated under bag semantics and return multisets (bags) as answers, since duplicates are not eliminated unless explicitly specified. The exact complexity of the query-containment problem for conjunctive queries under bag semantics has been an outstanding problem for more than twenty-five years. To this date, it is not even known whether conjunctive-query containment under bag semantics is decidable. The aim of this talk is to present a comprehensive overview of results about the query-containment problem for conjunctive queries and their variants under bag semantics, including recent results that reveal tight connections between this problem and open problems in information theory. Video

Wed 23.9.2020, 12-14  in Zoom (Meeting ID: 651 5733 7216, Passcode: 411213)
Jarkko Savela: Finding Periodic Apartments: A Computational Study of Hyperbolic Buildings

I will present a computational study of a fundamental open conjecture in geometric group theory using an intricate combination of Boolean satisfiability and orderly generation. Gromov’s subgroup conjecture (GSC) states that “each one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed surface of genus at least 2”. Non-right-angled groups are a class of groups whose status with regard to GSC is unknown, and, with this in mind, Kangaslampi and Vdovina constructed 23 such groups (via the theory of hyperbolic buildings) and analyzed their surface subgroups of genus 2 arising from periodic apartments. By developing novel SAT encodings and a specialized orderly algorithm we are able to extend the analysis to genus 3 and further to genus 4 utilizing parallel computation. Additionally, our work provides an independent verification of the genus 2 results of Kangaslampi and Vdovina.

Wednesday 30.09.2020  
Simon Blackburn and Cheryl Misak 14:00-16:00. SEMINAR BEGINS AT 14:00 SHARP

Cheryl Misak: 

The theory of general relativity drove Russell in 1928 to argue that we can refer to unobservable theoretical entities only through an understanding of their structural properties. At the end of that decade, two eminent philosophically inclined Cambridge mathematicians explored the issue. Simon Blackburn will show how Max Newman exploded Russell’s structuralism by noting that to say of two collections that they share a specified structureasserts nothing more than that they have the same cardinality. He will also show that Frank Ramsey is thought to have developed a technique (“Ramsey Sentences”) for the empiricist who wants to reduce theory to observation. Ramsey’s technique however, seems to open him to Newman’s problem, and Simon puzzles over why this seems not to have bothered him. 

Cheryl Misak will then argue that Ramsey in fact is not open to Newman’s Problem. Ramsey Sentences are much richer and much more interesting, in that they are situated in a context of inquiry and allow for refinement and improvement.

Simon Blackburn: Why is Newman missing?”

Abstract: It is generally agreed that the idea of the Ramsey sentence of a theory has an origin in “Theories” written in note form in 1929, the last year of Ramsey’s productive life. Yet in 1928 his friend Max Newman had published, in Mind, a paper which has ever since dominated discussions of Ramsification. The paper was directed at Russell’s 1927 book The Analysis of Mind, and Russell conceded its crticism was both fundamental and correct. Why then did Ramsey ignore it— when Russell had in effect preceded him in the application of Ramsey sentences in defining “structural realism” ? I suggest that the answer is that Ramsey was not interested in anything like Russelll’s foundational project (nor Carnap’s) but perhaps in something more like David Lewis’s 1970 paper “How to Define Theoretical terms”.  

SEMINAR CANCELLED: Wednesday 07.10.2020 12-14 Matteo Viale

Title: Tameness for set theory
Abstract: 
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.


Wednesday 14.10.2020 Miguel Moreno in Zoom (Meeting ID: 684 3587 2772, Passcode: 110592)

Title: Filter Reflection and Generalised Descriptive Set Theory

Abstract:Filter reflection is an abstract version of stationary reflection motivated from many results in generalised descriptive set theory. In this talk we will define filter reflection and different avatars of it. We will focus on its consequences in generalised descriptive set theory. We will also discuss how to force filter reflection and how to force the failure of filter reflection.
This is a joint work with Gabriel Fernandes and Assaf Rinot.


Wednesday 28.10.2020 Yurii Khomskii (Amsterdam University College and Universität Hamburg)
Title: Bounded Symbiosis and Upwards Reflection SLIDESKhomskii Helsinki 2020 with annotation.pdf

Abstract:

In [1], Bagaria and Väänänen developed a framework for studying the large cardinal strength of Löwenheim-Skolem theorems of strong logics using  the notion of Symbiosis (originally introduced by Väänänen in [2]). Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. We continue the systematic investigation of Symbiosis and apply it to upwards Löwenheim-Skolem theorems and upwards reflection principles. To achieve this, the notion of Symbiosis is adapted to what we call "Bounded Symbiosis".  As an application, we provide some upper and lower bounds for the large cardinal strength of upwards Löwenheim-Skolem principles of second order logic. 

This is joint work with Lorenzo Galeotti and Jouko Väänänen.

[1] Joan Bagaria and Jouko Väänänen, “On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals”, Journal of Symbolic Logic 81 (2) P. 584-604

[2] Jouko Väänänen, "Abstract logic and set theory. I. Definability.” In Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 391–421. North-Holland, Amsterdam-New York, 1979.

ZOOM info:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp


Wednesday 04.11.2020 12-14 Ralf Schindler 

Title: Martin's Maximum^++ implies the P_max axiom (*).

Abstract: 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."

Zoom info:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp

Wednesday 11.11.2020 12-14 Jeffrey Schatz 


Title: Axiom Selection after Large Cardinals: Maximize and the Question of CH Link to slides: After Large Cardinals

Abstract: There are two noted mathematical programs providing axioms extending the theory of Zermelo-Fraenkel set theory with large cardinals: the inner model program and the forcing axiom program. While these programs historically developed to serve different mathematical goals and ends, proponents of each have attempted to justify their preferred axiom candidate on the basis of its supposed maximization potential. Since 'maximize' proves central to the justification of ZFC+LCs itself, and shows up centrally in the current debate over how to best extend this theory, any attempt to resolve this debate will need to investigate the relationship between maximization notions and the candidates for a strong theory of sets. This talk will survey this project, discussing the history of 'maximization' considerations in set theory, introducing the main candidates for extending ZFC+LCs, and conclude by presenting recent results toward a resolution of these questions.

Zoom info:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp

Wednesday 18.11.2020 12-14 Matteo Viale

Title: Tameness for set theory 

Abstract: 
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.


Zoom info:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp


Wednesday 25.11.2020 12-14 Carolin Antos-Kuby 

Title: Two aspects of explanatoriness

Abstract: The phenomenon of explanation in mathematics is an interesting one: If there are different proofs for one theorem, all of them show that the theorem holds but often only some also show why the theorem holds, i.e. additionally also explain the theorem. Unlike in the natural sciences this phenomenon is not easily reducible to the phenomenon of causation. It is even unclear if there is only one form of explanatoriness or if it is a pluralistic notion. Here we give an example from recent descriptive set theory where we study two approaches to proving set-theoretic dichotomy theorems. We will see that both approaches provide explanations for the theorem, albeit in very different ways. We will use this to highlight two ways in which explanatoriness can be spelled out and distinguish between agent-dependent and agent-independent notions of explanatoriness. 

Zoom info:


https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp


Wednesday 02.12.2020 12-14 Miika Hannula

Title: On the Complexity of Horn and Krom Fragments of Second-Order Boolean Logic

Abstract: Dependency quantified Boolean formulae (DQBF) provide a canonical complete problem for non-deterministic exponential time. Restricted to Horn formulae this problem, then known as DQHORN, collapses to standard Horn satisfiability and is thus solvable in polynomial time. In this talk we isolate DQHORN as a fragment of second-order Boolean logic over CNF formulae with restrictions on clause and term structure. In particular, we consider whether some of the restrictions inherited from DQHORN can be loosened without increase in complexity. As a result, we obtain a complexity classification for various fragments of second-order Boolean logic.

Zoom info:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp


Wednesday 09.12.2020 12-14 Maria Aloni

Title: Pragmatic enrichments in team-based modal logic

Abstract: In a team-based modal logic, formulas are interpreted with respect to sets of possible worlds in a Kripke model rather than single worlds. In the talk I will present a bilateral version of a team-based modal logic with the non-emptiness atom NE (Yang and Väänänen 2017), which is motivated by linguistic phenomena at the semantics-pragmatics interface (including epistemic contradictions, free choice and ignorance inference).

Zoom info:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp


Wednesday 16.12.2020 17-18:30 Anand Pillay NOTE THE CHANGE OF TIME (and we start at 17:15)

Title:  Some model theory of fields.

Abstract. I will discuss various aspects of the study of fields using model-theoretic methods, as well as applications. Then I will discuss some 
recent work with Erik Walsberg on  the structure of fields which are "large" in the sense of Pop, and whose first order theory is simple (or stable).

ZOOM INFO:

https://us02web.zoom.us/j/4762106037?pwd=ckc1UzhDSHJmQ3I2bEpmNjNWcDNsZz09

Meeting ID: 476 210 6037
Passcode: HLGrp


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