# Papers

Below is an expanded list of papers. It includes a summary, slides (if they exist), commentary on the paper and more recent results, and corrections (if they exist).

My thesis:
• Will Boney, Advances in Classification Theory for Abstract Elementary Classes, PhD Thesis, 2014, pdf, 162 pages.
This is my thesis. At this point, all results have appeared (or will appear soon) in other and better places, so there's little value in looking at it (unless one wants to see how much worse early drafts can be). The better places to look are:
• Chapter III is Computing the Number of Types of Infinite Length
• Chapter IV is Tameness from Large Cardinal Axioms
• Chapter V is Forking in Short and Tame Abstract Elementary Classes
• Chapter VI is Tameness and Extending Frames
• Chapter VII is A Presentation Theorem for Continuous Logic and Metric Abstract Elementary Classes
• Chapter VIII is The \Gamma-Ultraproduct and Averageable Classes
The remaining parts are Chapter I "Introduction" (which might be an interesting overview of AECs), Chapter II "Preliminaries" (which might be useful as a quick tour through the basic definitions of AECs), and Chapter IX (which is hopefully coming together soon).

My papers: Published (and accepted) papers
1. Will Boney, Tameness from Large Cardinal Axioms, Journal of Symbolic Logic, vol 79, no 4, Dec 2014, 1092-1119. publisher version pdf arXiv
This paper derives various locality properties of Galois types from large cardinals. It uses this to, in particular, prove that Shelah's Categoricity Conjecture follows from class many strong compacts.
Slides of a talk based on this paper can be found here.
The final section "Further Work" is made obsolete by the below paper Large Cardinal Axioms from Tameness in AECs (joint with Spencer Unver). In particular, many of the implications in this paper are in fact equivalences when almost'' is added to the large cardinal.
There is a typo in Definition 2.3.(6): it should read "...with LS(K) = \lambda..."

2. Will Boney and Rami Grossberg, Forking in Short and Tame Abstract Elementary Classes, Annals of Pure and Applied Logic, vol 168, no 8, 2017, 1517-1551. publisher version pdf arXiv
This paper introduces \kappa-coheir as an independence relation in short and tame AECs. The main goal is to distill the model-theoretic content of Makkai-Shelah and transfer it to genereal AECs. This is done and a notion of \kappa-heir is introduced. The U-rank is developed in greater generality and large cardinals are shown to make everything easier.
Slides of a talk based on this paper can be found here. A poster based on this paper can be found here. This is a picture of me next to the poster at the Model Theory 2013 meeting in Ravello, Italy.

3. Will Boney, Tameness and Extending Frames, Journal of Mathematical Logic, vol 14, no 2, 2014. publisher version pdf arXiv
This paper combines good frames and tameness and shows that a good \lambda-frame plus \lambda-tameness for 1- and 2-types implies the existence of a good \geq \lambda-frame. A detailed proof is given of Shelah's result that a good \lambda-frame implies the uniqueness of limit models in \lambda.
It has since been shown that the use of tameness for 2-types is unnecessary by myself and Vasey; and Jarden (independently). See Boney and Vasey, Tameness and Frames Revisited for more details.

4. Will Boney, Computing the Number of Types of Infinite Length, Notre Dame Journal of Formal Logic, vol 58, no 1, 2017, 133-154, publisher version pdf arXiv, 22 pages
This paper computes the supremum of the number of types of infinite length over models of a fixed size from the supremum for the number of one types. No such computation is possible for nonalgebraic types, but we introduce a generalization (strongly separative types) that admits an upper bound.

5. Will Boney, Rami Grossberg, Alexei Kolesnikov, and Sebastien Vasey. Canonical Forking in AECs, Annals of Pure and Applied Logic vol 167, no 7, 2016, 590-613, publisher version, pdf arXiv
This paper explores nonforking relations in AECs axiomatically and characterizes when a nonforking is canonical. These results are applied to coheir from Forking in Short and Tame Abstract Elementary Classes and Shelah's good \lambda-frames.

6. Will Boney and Sebastien Vasey, Tameness and Frames Revisited, Journal of Symbolic Logic, vol 82, no 3, 2017, 995--1021, publisher version, pdf arXiv.
This paper continues Tameness and Extending Frames above and removes the assumption of tameness for 2-types. It does so by analyzing when properties transfer to frames of longer tuples.

7. Will Boney, A Presentation Theorem for Continuous Logic and Metric Abstract Elementary Classes, Accepted, Mathematical Logic Quarterly, pdf arXiv, 27 pages (Updated 11/1/2016).
This paper shows that continuous first order logic can be viewed as a particular fragment of L_{\omega_1, \omega} by analyzing the dense subsets of the models. This analysis is extended to MAECs by showing that they can be analyzed as (discrete) AECs.
A previous iteration had a separate version with some proofs, but these have now been restored to a single version.

8. Will Boney and Sebastien Vasey, Chains of Saturated Models in AECs, Archive for Mathematical Logic, vol 56, no 3, 2017, 187-213, publisher version pdf arXiv
This paper generalizes the first-order fact that, in nice enough theories, the union of \lambda-saturated models are \lambda-saturated to AECs. Two main methods are used: independence relations and averages of types.

9. Will Boney and Spencer Unger, Large Cardinal Axioms from Tameness in AECs, Proceedings of the American Mathematical Society, vol. 145, no 10, 2017, 4517-4532, publisher version pdf arXiv.
Building on ideas of Shelah's Maximal failure of sequence locality in AECs, we give converses to results of Tameness from Large Cardinal Axioms by adding almost'' to the large cardinal properties (the importance of , e. g., almost strongly compact cardinals was also discovered by Brooke-Taylor and Rosicky in Accessible Images Revisited). This (and the work of Brooke-Taylor and Rosicky) allow us to prove a triple equivalence between a) the existence of class many strongly compact cardinals, b) the statement that all AECs are eventually tame, and c) a category-theoretic principle originally isolated by Makkai and Pare.

10. Will Boney, Rami Grossberg, Michael Lieberman, Jiri Rosicky and Sebastien Vasey. \mu-Abstract Elementary Classes and other generalizations, Jornal of Pure and Applied Algebra, vol 220, issue 9, Sep 2016, 3048-3066, publisher version pdf arXiv
We explore the natural generalization of AECs that can capture L_{\lambda, \mu}, called \mu-AECs. We generalize some theorems to them and prove that \mu-AECs are equivalent (as categories) to accessible categories.

11. John Baldwin and Will Boney, Hanf Numbers and Presentation Theorems in AECs, Beyond First Order Model Theory (José Iovino ed.), CRC Press (2017), 327–352. pdf arXiv
This paper has two main goals: first, we prove that a strongly compact cardinal is an upper bound on the Hanf number for amalgamation (and related results) by giving syntactic characterizations. Second, we give a relational presentation theorem that is canonical/functorial (as opposed to Shelah's) in order to do the above without disjointness.

12. Will Boney and Sebastien Vasey, A Survey on Tame Abstract Elementary Classes, Beyond First Order Model Theory (José Iovino ed.), CRC Press (2017), 353–427. pdf arXiv
This is a survey on current (as of writing) results in tame AECs. As a motivation/example of the techniques, we outline a proof of Vasey's categoricity result in universal classes.

13. Will Boney and Sebastien Vasey, Good Frames in the Hart-Shelah Example, Accepted, Archive for Mathematical Logic, pdf arXiv, 24 pages (Updated 11/08/17).
We analyze the frame properties of the good frames in the Hart-Shelah example that were isolated in Tameness and Extending Frames, Section 10. We show that the maximal good frame there is type-full, but fails the existence property for uniqueness triples.

14. Will Boney, Rami Grossberg, Monica VanDieren, and Sebastien Vasey, Superstability from Categoricity in Abstract Elementary Classes, Annals of Pure and Applied Logic, vol 168, no 7, 1383-1395, publisher version pdf arXiv
We generalize the "categoricity implies superstability" result from [ShVi635] to arbitrary independence relations and isolate the ZFC portions of it.

Submitted Papers
1. Will Boney and Monica VanDieren, Limit Models in Strictly Stable Abstract Elementary Classes, Submitted, pdf arXiv, 18 pages (Updated 2/23/16).
We look at uniqueness of limit models in strictly stable AECs. The results suggest that "For which \alpha do we have (\mu, \alpha)-limit models are (\mu, \mu)-limit models?" as a potential dividing line for AECs.

2. Will Boney, The \Gamma-ultraproduct and averageable classes, Submitted, pdf arXiv, 26 pages (Updated 8/16/17)
This paper looks for compactness results in classes that omit a set of unary types \Gamma by introducing a suitable ultraproduct (called the \Gamma-ultraproduct). It discusses examples when the \Gamma-ultraproduct is poorly behaved, sufficient conditions for good behavior, and examples of these conditions (with an extended look at torsion modules over PIDs). See also "Coheir in Averageable Classes" below.

3. Will Boney, Definable Coherent Ultrapowers and Elementary Extensions, Submitted, pdf arXiv, 14 pages (Updated 8/20/17)
We cross the method of extenders from set theory with the notion of definable ultrapowers to develop definable coherent ultrapowers. This allows us to characterize any extension in an L_{\infty, \omega}-class with definable Skolem functions as a coherent ultrafilter on the base.

4. Will Boney, Model Theoretic Characterizations of Large Cardinals, Submitted, pdf arXiv, 25 pages (Updated 8/24/17)
We extend previous results (some folklore, some due to Magidor and especially Benda) that characterize large cardinals in terms of model theoretic compactness. We prove general frameworks to pin down the exact large cardinal strength of different compactness for omitting types properties of infinitary, second-order, and sort logic.

5. Will Boney and Sebastien Vasey, Structural Logic and Abstract Elementary Classes with Intersections, Submitted, pdf arXiv, 14 pages (Updated 1/13/18).

6. Nathanael Ackerman, Will Boney and Sebastien Vasey, Categoricity in Multiuniversal Classes, Submitted, pdf arXiv, 16 pages (Updated 4/24/18).

7. Will Boney and Ioannis Souldatos, A Lower Bound for the Hanf Number for Joint Embedding, Submitted, pdf arXiv, 17 pages (Updated 8/9/18).

8. Will Boney, Erdős-Rado Classes, Submitted, pdf arXiv, 23 pages (Updated 10/2/18)

Papers in Preparation
1. Will Boney and Sebastien Vasey, Categoricity and Infinitary Logics, Preprint, pdf arXiv, 9 pages (Updated 10/26/15).
The original goal was to fix a gap in Conclusion IV.2.14 in Shelah's Classification Theory for Abstract Elementary Classes and an exploration of model-theoretic forcing in AECs. However, we identified a gap in our gap filling, so this is what remains.

2. Will Boney and Pedro Zambrano, Around the set-theoretical consistency of d-tameness of Metric Abstract Elementary Classes, Preprint, pdf arXiv, 9 pages (Updated 8/22/15).
We extend the results of Tameness from Large Cardinal Axioms to Metric AECs by way of the functor introduced in A Presentation Theorem for Continuous Logic and Metric Abstract Elementary Classes.

3. Will Boney, No Maximal Models from Looking Down, Preprint, pdf arXiv, 7 pages (Updated 11/2/15)
This paper summarizes Shelah's AECs with with not too many models and replaces this assumption with amalgamation below \lambda.

Below are some informal notes I've written, along with commentary. These are very informal and I welcome any comments or correction.
• Coheir in Averageable Classes, pdf
This is a supplement to The \Gamma-ultraproduct and averageable classes that goes through the details of generalizing Forking in Short and Tame Abstract Elementary Classes to the context of averageable classes.

• Shelah's Omitting Types Theorem in Excruciating Detail, pdf
In [Sh394], Shelah introduces and proves an omitting types theorem for AECs that further generalizes Morley's. This is really amazing, but the proof appearing there was not very satisfying (nor was the proof appearing in Baldwin's book). So I've written up a proof in extreme detail.

• Feferman-Vaught Theorem, pdf
The Feferman-Vaught Theorem is to reduced products (products modulo a filter) what Los' Theorem is to ultraproducts. This is a write up of the proof based on the one given in Chang-Keisler. I had once hoped that the ZFC existence of \kappa-complete filters and this theorem would give nice nonelementary compactness results. As indicated in the note, this turned out not to be the case.

• Zilber's Pseudoexponentiation, pdf
This is a write up of talks I presented to the UIC Model Theory Seminar in the Fall of 2014 on Zilber's pseudoexponential fields. It glosses over many details and belabors others, but might be useful for those (like myself in Fall of 2014) with a weak background in algebraic geometry. It benefits from the helpful comments of John Baldwin, Dave Marker, and Jonathan Kirby.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1402191.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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