# Teaching

Information regarding the course Model Theory of differential fields in characteristic zero.

## Practical information

The lectures take place on the Fridays from 9.45am to 11.15am and from 11.30am to 1.00 pm with a fifteen minutes break in the middle. The rooms for each lecture are indicated below. Be careful the room change with the lectures.

- Lectures 1/2 on 12/01/2024. Quai 43, s.104 (1er étage)
- Lectures 2/3 on 19/01/2024. Darwin D71 (RDC)
- Lectures 3 on 26/01/2024. Braconnier Séminaire 1
- Lectures 4/5 on 02/02/2024. Darwin C salle Caullery (RDC)
- Lectures 5/6 on 09/02/2024. Darwin C salle Caullery (RDC)
- Lectures 6/7 on 23/02/2024. Darwin C salle Caullery (RDC)
- Lectures 8 on 08/03/2024. Braconnier Séminaire 1
- Lectures 9 on 15/03/2024. Braconnier Séminaire 1

More information about the location of each of the rooms can be found here .

## Lecture notes

- Lecture 1. Motivation
- Lecture 2. Ritt-Raudenbush theory
- Lecture 3. Differentially closed fields
- Lecture 4. Seidenberg embedding Theorem
- Lecture 5. Differential forms
- Homework. Homework due on March 8th
- Lecture 6. The Ax-Schanuel Theorem

## References

- David Marker, Model Theory of differential fields (second chapter of the book Model Theory of fields) pdf.
- Rosenlicht, On Liouville theory of elementary functions, Pacific Journal of Mathematics, Vol. 65, No. 2, 1976. pdf.
- Bouscaren E., Model Theory and Algebraic Geometry, Lecture notes in Mathematics link .

## Oral examination

For the oral examination, you will have to choose and present one of the following subject (out of three randomly picked subjects) in the following list

- Prime differential ideals of differential polynomials (Lecture 2)
- Radical differential ideals of differential polynomials (Lecture 2)
- Elimination of quantifiers in the theory of differentially closed fields and consequences (Lecture 3)
- Elimination of imaginaries in the theory of differentially closed fields (Lecture 3)
- The Seidenberg embedding Theorem (Lecture 4)
- Algebraic properties of differential forms (Lecture 5)
- The Galois-theoretic proof of the functional Lindermann-Weierstrass Theorem (Lecture 1)
- The Liouville-Rosenlicht Theorem (Lecture 6)
- The Ax-Schanuel Theorem (Lecture 6)

The oral examinations will take place on Tuesday March 26th from 1pm to 6.30pm in room Berthollet 126 (1er étage) based on the following schedule.

- 1.15pm to 2pm: Youssef
- 2pm to 2.45pm: Alexandre
- 2.45pm to 3.30pm: Yoann
- 3.30pm to 4.15pm: Enzo
- 4.15pm to 5pm: Quang-Khai

Please come 10 minutes before the start of your oral exam.