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Calendar for Spring 2025
The model theory seminar meets Wednesdays, 3-4pm (with the possibility of short extension) in 732 Evans. This semester, participants will present their own recent research or they will expose interesting new results from the literature. Wednesday, January 21st, 2026: … Continue reading
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Amalgamation
Let $T$ be a theory with a good notion of independence. Generally, we have in mind forking independence, though something weaker would be acceptable. In fact, for our proof, the only properties we use are that independence is invariant under … Continue reading
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ACFA quantifier simplification
Our proof of model completeness for ACFA yields strong quantifier simplification results. Let us start with a relative completeness result. If a theory $T$ is model complete, then whenever $E \models T$ and $E \subseteq M_i \models T$ for $i … Continue reading
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Axiomatizing ACFA
The theory of difference fields admits a model companion, ACFA, often called “algebraically closed fields with a generic automorphism”. To prove this, we need to show that the class of existentially closed difference fields is first-order axiomatizable. We have … Continue reading
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Calendar for Autumn 2025
This is an archived version of the calendar for the Autumn 2025 seminar. Four people have volunteered to speak about specific topics. The dates of their lectures will be announced later. Harper Wells will speak about the problem of the … Continue reading
Introduction to the model theory of difference fields
Before we get to the scientfic content, allow me to remind you that the location for the seminar has changed to 891 Evans and we meet Wednesdays, 3-4pm. Definition: A difference field $(K,\sigma)$ is a field given together with … Continue reading
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Autumn 2025 Model Theory Seminar on Model Theory of Difference Fields
Our seminar this semester will focus on the model theory of difference fields. We meet in 891 (note the change in location!) Evans, Wednesdays 3-4pm. The Course Control Number is 15443. The main sources are the following research papers. … Continue reading
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Archives of Academic Year 2024/5 seminars
Learning seminar on o-minimality (Autumn 2024) The learning seminar on o-minimality met on Tuesdays and Thursdays, 11am-12noon in 891 Evans during the Autumn 2024 semester. The Course Control Number was 34544. For the most part, we followed the presentation in … Continue reading
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O-minimal Euler characteristic
Every o-minimal structure admits an Euler characteristic with values in $\mathbb{Z}$. Our computation showing that $K_0(\mathfrak{R})$ is a quotient of $\mathbb{Z}$ when $\mathfrak{R}$ is an o-minimal expansion of an ordered field may be reversed to define the o-minimal Euler characteristic.
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The Grothendieck ring of an o-minimal expansion of an ordered field as a quotient of $\mathbb{Z}$
If $\mathfrak{R}$ is an o-minimal expansion of an ordered field, then $K_0(\mathfrak{R})$ is a quotient of $\mathbb{Z}$. In a later post, we will see that, in fact, $K_0(\mathfrak{R})$ is exactly $\mathbb{Z}$.
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