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This collection of notes results from a series of 8 lectures given by Dr Gareth Jones at the Fields Institute over the course of 19 Jan to 11 Feb 2022. The vast majority of the material presented here was given as part of this first module in the three-module course on \omy and its use in proving the \pwt, along with some applications. Primarily, an attempt was made to uniformize notation and formatting across the various lectures. Some extra details were added where otherwise missing or left as an exercise to the viewer -- or even more likely, of some confusion to this author.

This current iteration of the notes is to be considered preliminary, and the style and particularities of their presentation are by no means final. As the owner of the sole pair of eyes to have read this document, no assurances are made that it is free of mistakes or oversights (in fact, one might be more willing to make promises to the contrary) -- but to the best of our ability, we have tried to minimize the sure-to-come list of corrigenda (and perhaps if we're particularly unlucky) errata upon review by fresh eyes.
 
This document intends to act as an easily (to an early graduate student in mathematics) accessible introduction to \omy in the context of expansions of the real field and their application in proving the \pwt. In particular, \omy is not discussed in full generality -- rather, the role that \omy of ordered fields plays in results on cell decompositions and parameterization by these decompositions, and of course, how this all comes together to prove the eponymous theorem of Pila and Wilkie are discussed. The ordering of content presented is not strictly adherent to the delineation given in the lectures, but it does broadly follow.

As most things are, these notes are written from a particular authorial perspective -- which is to say, not by those with more than little to any background in model theory. There may be a belabouring of ideas that another would find trivial, unnecessary, or otherwise not necessarily worth the space they take up on the page. Compiling these notes is not just to archive the lecture series given but also to make it somewhat more accessible by clarifying what we ourselves had to investigate further and understand as we attended the course. This does not mean any significant amount of material is added above and beyond the course content itself -- rather, more so that some `one-liners' in the original presentation are afforded just a few more in these notes. Some contextualization was added in the first section to bridge the gap for a student for whom \omy is completely novel.

Where not otherwise cited, facts should be supposed to have been taken from the lectures — which themselves will periodically include references or suggestions for additional reading material. All citations are included for exteriorly sourced information, with the full list of references listed at the end of this chapter. 
%This is not necessarily congruent with the Springer Nature style of chapter-end citations. Still, the struggle with incorporating both this feature and BibTeX at the hands of this authorial team was nothing short of nightmarish. And, as with so many things in life, BibTeX should always come first.

Although at times dry, the content here, especially once one delves into it and splashes about in the waters of its intrigue, is quite the scene to behold. However, we would be remiss to assume this view is shared by one new to the subject area, or only acquainted with logic and not number theory, vice versa -- or (may the good Lord help you) neither. Some attempt at levity is made throughout to keep things interesting while still rigorous, complete, and unobtrusive. Appropriateness of this attempt is yet to be determined, but it is the philosophy of this author that texts are to be enjoyed -- perhaps even the otherwise driest more so than any other; how else can we be expected to pay sufficient attention unless we are sufficiently engaged, amused, and interested all at once. Perhaps this is just one author's opinion unshared by the mathematical community at large -- but either way, you, my reader dearest, have no choice in the matter and so shall have to decide for yourself the merit and suitability of the approach for the matters at hand. With all that in mind, please enjoy.
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%\fix{An aside for Dr Speissegger: throughout, there are bits of red text indicating the non-inclusion or reduction of detail from what should otherwise be in these notes. This is to clarify where some corners were cut in the interest of time -- as I tended to take a breadth-over-depth approach to put the course material into notes, feeling that would be a more `complete' version, despite being similarly unfinished either way.}

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Alun C. Stokes \hfill {\it McMaster University}\\
\monthyeardate\today \hfill {\it MATH 712}\\
%May 2022 \hfill {\it MATH 712}\\
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