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# Model theory and set theory days, May 20-26, 2016: 2 Lecture series by Boban Velickovic and Andres Villaveces.

I. Boban Velickovic: The tree property and related combinatorial principles

Abstract: Compactness of first order logic is one of the central paradigms in mathematics. When one tries to extend the compactness theorem to infinitary logics, one is lead to large cardinals such as  as weakly compact, strongly compact and supercompact cardinals.  In many applications one would like to transfer fragments of compactness for infinitary logic to small cardinals such as, for instance, the \aleph_n, for finite n. The tree property is one of the most fruitful way of achieving this transfer.  In this mini course we will survey some classical and some recent results on the tree property and related combinatorial statements. We will show that this properties may hold at various small cardinals (for instance, \aleph_n (for n \geq 2) or \aleph_{\omega+1}).  We will also present the main applications of the tree property. Finally, we will discuss the key open questions which are the subject of current research in this area.

## II. Andres Villaveces: Language, Logic and Nonelementary Classes: external/internal interactions.

Abstract: In this minicourse, I will explore three independent issues around the interaction between language and nonelementary classes: emergence and control (The Presentation Theorem for AECs), invariance and groups (the SIP property for homogeneous classes) and interpolation (comparing different AECs, lifting classical comparisons from Abstract Model Theory). The minicourse will be roughly self-contained (for people with some training in logic). It will combine classical parts with more recent constructions. The main focus is on the interaction between language and classes of structures (through logic), as highlighted by three different theorems. The course is not strongly cumulative: the three topics have independent motivations.
• The Presentation Theorem: this is a classical theorems due to Shelah, basic in AECs. I will give basic definitions, a statement of the theorem, a sketch of proof and various connections to other topics (I will describe its huge impact in the development of the Model Theory of AECs).
• The Small Index Property (SIP) for Homogeneous Classes: this recent work of mine (jointly with Zaniar Ghadernezhad) extends results of Lascar and Shelah to some AECs. I will explain the role of the SIP in understanding the "Reconstruction Problem" (when can one reconstruct a structure M from its symmetries Aut(M)). I will explore consequences and I will describe our setting for generalizing Lascar-Shelah to some non-elementary classes.
• Categories, AECs, Interpolation: this last construction intends to compare different AECs with tools analogous to those one uses to compare theories in classical settings - and a little bit of Abstract Model Theory (beginning). We start building some functors extracted from underlying languages and then use the Presentation Theorem to lift these functors to other functors between various AECs. This work explores a line different from that of generalizing AECs to concrete categories: we rather use a bit of category theory to understand some intereactions between classical AECs. This is joint work with Hugo Mariano and Pedro Zambrano.

The level will be kept basic and accessible, with occasional forays into more technical matters. The aim of the minicourse, in addition to presenting a perspective into "linguistic" aspects of AECs, will be to open room for "philosophical" aspects of AECs typically left aside in courses. The second and third topic have never been presented in the form of a minicourse, so in this sense this will be a "first run".

## SCHEDULE: ALL LECTURES ARE IN C122 (Exactum, Kumpula)

Friday May 20

14-16 Andrés Villaveces

#### Monday May 23

10-12 Boban Velickovic14-16 Andrés Villaveces

#### Tuesday May 24

10-12 Boban Velickovic, 14-16 Andrés Villaveces

#### Wednesday May 25

10-12 Boban Velickovic

#### Thursday May 26

10-12 Boban Velickovic

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