Math 512:

o-minimality and applications

Math 512 is a graduate topics course. We meet MWF at 2:00 PM on zoom.

The password is the city in which the above pictured campus resides followed by n, where n is the natural number for which: "a theory can not have exactly n countable models".

Office hours: M, Tu, W, Th 1-2 PM via the same Zoom link as class. Additional office hours are completely possible - please email to schedule.

The subject of o-minimality emerged in the 1980s through the work of models theorists. It is a natural generalization of real algebraic and analytic geometry to a certain abstract framework. I like to think of work on the subject as roughly fitting into three general areas.

Developing a general structure theory for definable sets in o-minimal structures, e.g. cell decomposition and curve selection.
Establishing that particular structures are o-minimal and proving results about the structures which could be o-minimal or about other tameness results for expansions of o-minimal structures. Perhaps the most important examples of o-minimal structures are the real field, the real field expanded by the exponential function, the real field expanded by restricted analytic functions, and the combination of these.
Applications of o-minimality - we will concentrate on what is now the extremely active area of number theoretic applications.

Most of the results from the first lecture which we didn't prove will be returned to and given a proof in the first couple of weeks (e.g. the Monotonicity theorem).

Part I: The basics of o-minimality

The notes and references will appear in the dropbox link for part 1.

These notes might actually contain a bit more model theory than subsequent upcoming parts, but I do try to include a basic overview, and I also try to indicate in the notes any point where I am using some model theoretic result which those who haven't studied the subject are unlikely to know.

I am writing my own notes, but the standard reference on the subject is the book of van den Dries pictured here. The notes in the dropbox will change as I write them.

Part 2: Cell decomposition

The notes and references in the linked dropbox all center around results which lead to a proof of the Cell Decomposition Theorem.

Part 3: Analytic functions

We've seen that the real numbers as a field give rise to an o-minimal structure in which the definable sets are called semi-algebraic. Following this, we developed strong structural consequences of this notion of o-minimality (cell decomposition, curve selection, etc.).

There is still hard work required when you want to prove that a given structure is o-minimal, and in this part we are going to see how to do this when we consider analytic functions with restricted domains. See the dropbox for notes.

Part 4: Number theory

In the early 2000s, Wilkie noticed that o-minimality gave strong implications with regard to the distribution of rational points on definable sets. Since then there has been an incredible expansion of results which draw number theoretic conclusions from o-minimal tools. We will cover the Pila-Wilkie Theorem in detail in this part. See the dropbox for notes.

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