This course is appropriate for people with basic knowledge of abstract algebra and commutative algebra. Some knowledge of differential geometry, mathematical logic or some number theory is welcome, but not required.
A DMATH forum is available for discussion and to ask/answer questions about the course, see here.
The lecture will be accompanied by roughly biweekly exercise classes, usually during the Thursday class. We will announce the precise dates in the lecture as well as here. You should submit your exercise sheets either in person in the Monday lecture before the next exercise class, or in the designated box in HG J 68, or in PDF form to the coordinator: raphael.appenzeller [at] math.ethz.ch.
| Dates of exercise classes |
|---|
| September 29, October 6, October 20 |
| Exercise sheet | Due by | Solutions |
|---|---|---|
| Exercise sheet 1 | Mo, October 3 | Solutions 1 |
| Exercise sheet 2 | Mo, October 17 | Solutions 2 |
| Exercise sheet 3 | Mo, October 31 | Solutions 3 |
| Exercise sheet 4 | Mo, November 21 | Solutions 4 |
| Exercise sheet 5 | Mo, December 5 | Solutions 5 |
| Exercise sheet 6 | Mo, December 19 | Solutions 6 |
| Day | Content |
|---|---|
| 22.9.2022 |
Introduction to o-minimality; outline of the content of the
course. Basic definitions of model theory (language,
structure, terms, formulas, definable sets).
Chapter 1 (introduction) Chapter 2 (model theory), beginning |
| 26.9.2022 |
Basic model theory, continuation (theories, models,
etc).
Chapter 2 (model theory; corrected mistake on definition of embedding) |
| 29.9.2022 | Exercise class |
| 3.10.2022 |
The compactness theorem (statement, examples, beginning of
proof).
Chapter 3 (the compactness theorem). |
| 6.10.2022 | Exercise class |
| 10.10.2022 |
End of proof of the compactness theorem.
Chapter 3 (the compactness theorem). Quantifier elimination: definition, examples, model-theoretic criterion. Chapter 4 (quantifier elimination, beginning). |
| 13.10.2022 |
Proof of the model-theoretic criterion for q.e.. Proof of
q.e. for Real Closed Fields.
Chapter 4 (quantifier elimination). |
| 17.10.2022 |
Simple consequences of q.e. for real closed fields.
Chapter 4 (quantifier elimination). O-minimal structures: elementary topological facts. Chapter 5 (beginning) |
| 20.10.2022 | Exercise class |
| 24.10.2022 |
O-minimal structures: selection principle, monotonicity
theorem.
Chapter 5 (beginning) |
| 27.10.2022 |
O-minimal structures: monotonicity
theorem (end of proof), uniform finiteness theorem.
Chapter 5 |
| 31.10.2022 |
O-minimal structures: end of the proof of the uniform
finiteness theorem.
Chapter 5 Cellular decomposition theorem: statement and elementary properties of cells. Chapter 6 |
| 3.11.2022 |
Cellular decomposition theorem and corollaries
Chapter 6 |
| 7.11.2022 | No class |
| 10.11.2022 | Exercise class |
| 14.11.2022 |
Cellular decomposition theorem: beginning of the proof.
Chapter 6 |
| 17.11.2022 |
Cellular decomposition theorem: end of the
proof.
Chapter 6 |
| 21.11.2022 |
Dimension theory for definable sets.
Chapter 7 (corrected) |
| 24.11.2022 | Exercise class. |
| 28.11.2022 |
(Arithmetic) applications of o-minimality: quick survey;
statement of the Pila-Wilkie Theorem.
Chapter 8 (beginning) |
| 1.12.2022 |
First steps in the proof of the Pila-Wilkie Theorem: the
"determinant method".
Chapter 8 (beginning) |
| 5.12.2022 |
End of the proof of the Pila-Wilkie Theorem, assuming
existence of reparameterizations.
Chapter 8 (beginning) |
| 8.12.2022 | Exercise class. |
| 12.12.2022 |
Start of the proof of Laurent's Theorem using the
Pila-Zannier strategy.
Chapter 8 (beginning, corrected) |
| 8.12.2022 |
End of the proof of Laurent's Theorem.
Chapter 8 (beginning) |