Special Session: Model Theory and Interactions

Session description

Our goal in this special session is to bring together a group of researchers who share a background of model theory, plus a little extra: we will explore model theory through its interactions within and without the boundaries of mathematical logic. The talks we hold will introduce novel model-theoretic applications to researchers in logic, as well as possibly introduce the model-theoretic domain as a whole to a host of researchers gathering or working at Cal Poly. 

Location & Dates

An interactive map of Cal Poly is available HERE!

Parking: Visit the parking webpages for information and the locations of lots. There are fees for parking. Participants may use 131 Parking Structure and 130 GS Grand Parking, both off Grand Avenue on Pacheco Way.

Meeting App: Download the meeting app for the latest information.

Room Locations On-site: The registration desk (+ badge pickup + coffee service) will be in the CAFES Pavilion (Building 19A). The registration desk will be open on Saturday, May 3, from 7:00 am to 4:00 pm and Sunday, May 4, from 7:30 am to noon. Invited Addresses will be in Room E27, in Building 52. The Model Theory and Interactions special session will be in Building 38 (room 202) — same building as the Logic and Analysis special session (room 204). The poster session and reception will be held in the CAFES Pavilion (Building 19A). 

Wi-Fi: Cal Poly uses Eduroam as the primary secure wireless network. If your school also uses Eduroam, you should be able to connect to the same internet. Alternatively, Cal Poly has a wireless network for all campus guests that is broadcast as "CalPolyGuest." To connect to the network, you will simply need to click the "Login" button, and before you can start browsing, you will have to accept the terms of use.

Lunch: 1901 Marketplace is next door to building 19A (registration) and has a Chick-Fill-A and a salad bar. Next door there is a Starbucks, while a short walk away is a Scout coffee and Market Grand Avenue (a deli and a cafe). The full list of campus dining options is available HERE. There are a few dining options in town too. 

Contact me for further information concerning travel, accommodation, and dining recommendations.

In addition to Model Theory and Interactions, a special session on Logic and Analysis will take place concurrently. We encourage participants to review its schedule as well.

Purity and chains of modules.  Dr Soinbhe Nic Dhonncha, Cal Poly.

The notions of purity and pure-injectivity are central to the model theoretic study of modules. As the study of modules, and of the category of modules over a fixed ring, has extended to the study of module categories (certain functor categories), the notions of purity and pure-injectivity have also extended naturally to these contexts. One module category of interest is the category whose objects are integer indexed chains of modules over a fixed ring R, which we denote by ModRA_{\infty}^{\infty}. In this talk, we discuss the relationship between the notions of purity and pure-injectivity in ModRA_{\infty}^{\infty}, and of purity and pure-injectivity in the category ModR.

Expansions of the group of integers by an algorithmically random set.  Dr Michael Hehmann.

In investigating the number of non-isomorphic models of a first-order theory in a given cardinality, Saharon Shelah developed a number of "dividing lines" which serve to classify theories based on whether or not models of the theory omit certain combinatorial complexity. Recent work has been done to classify theories of expansions of the additive group of integers with respect to these combinatorial dividing lines. Following on Kaplan & Shelah (2016) giving a classification result for the integers expanded by a predicate for the prime numbers, Bhardwaj & Tran (2017) proved an analogous result for the expansion by the square-free numbers, and asked if this classification result could be proved for other "sufficiently random" sets. In this talk, I show how to give one answer to this question by leveraging notions of algorithmic randomness from computability theory.

On f-generic types in NIP groups. Atticus Stonestrom, University of Notre Dame.

Recall that a definable group is said to be ‘definably amenable’ if it admits a Keisler measure invariant under left-translation. Definable amenability has proved a key hypothesis in much of the generalization of stable group theory to the more general setting of NIP groups, and an important role in this structure theory is played by ‘f-generic’ types, which serve as the appropriate analogues of ‘generic’ types from the stable setting. I will define and discuss these notions, and then present a technical result: for an NIP group, definable amenability is equivalent to the existence of an f-generic type. This positively answers a question from Chernikov and Simon. As a quick application, I will show that every dp-minimal group is definably amenable.

The challenge in capturing classes of structures in continuous logic. Professor Isaac Goldbring, UC Irvine.

With a few exceptions, treating various algebraic and combinatorial structures as models of a first-order theory is often fairly routine. In contrast, capturing natural classes of structures from analysis as models of a first-order continuous theory can often be quite challenging. After presenting some simple examples which nevertheless illustrate the issue at hand, we discuss a couple of more complicated examples, namely unitary representations of locally compact groups and W*-probability spaces. The former represents joint work with Itaï Ben-Yaacov and the latter joint work with Jananan Arulseelan, Bradd Hart, and Thomas Sinclair.

A theory which really doesn’t have a computable model. Dr Patrick Lutz, UC Berkeley.

The completeness theorem guarantees that any consistent theory has a model, but it is not hard to show that the computable version of this statement does not hold: there is a computable, consistent theory with no computable model. Furthermore, there are many natural theories with this property. In particular, the proof of Tennenbaum’s Theorem can be used to show that essentially any reasonably strong set theory, including ZFC, cannot have a computable model. Recently, however, Pakhomov showed that this phenomenon is somewhat fragile: it depends on the language in which the theory is formulated. For example, Pakhomov showed that there is a theory which is definitionally equivalent to ZFC (a strong form of bi-interpretability) which does have a computable model. In light of this, Pakhomov raised the question of whether every computable, consistent theory is definitionally equivalent to a theory with a computable model. In response to Pakhomov’s question, James Walsh and I have constructed a counterexample: a computable, consistent theory such that no definitionally equivalent theory has a computable model. I will explain the context for our result and the main ingredients of the proof, which consist of a mixture of computability theory and tame model theory.

Transcendence results via model theory. Professor James Freitag, UIC.

In recent years, a number of advances in model theory have lead to proofs of new functional transcendence results. In this talk, we will introduce a few of the methods and talk about the prospects for generalizing the existing results.

Abstract Group Chunks. Ronan O'Gorman, UC Berkeley.

The group chunk theorem, and its extension the group configuration theorem, are key tools in geometric stability theory. I will discuss ongoing work on how, using the language of presheaves on sites, these results can be generalized beyond the model-theoretic context, with hopeful applications to algebraic geometry.

A Borovik-Cherlin bound for primitive pseudo-finite permutation groups. Professor Nick Ramsey, University of Notre Dame.

A primitive permutation group (X,G) is a group G together with an action of G on X such that there are no nontrivial equivalence relations on X preserved by G.  An rough classification of primitive permutation groups of finite Morley rank, modeled on the O'Nan-Scott theorem for finite primitive permutation groups, has been carried out by Macpherson and Pillay and this classification was then used by Borovik and Cherlin to prove that if (X,G) is a primitive permutation group of finite Morley rank, the rank of G can be bounded in terms of the rank of X.  We study the analogous situation for pseudo-finite primitive permutation groups of finite SU-rank, building both on supersimple group theory and classification results of Liebeck-Macpherson-Tent.  This is joint work with Ulla Karhumäki.